As long as the execution method is not any more severe than Biblical stoning and burning, so that the man or woman is intentionally made to suffer through something like unjust corporal punishment or flaying or starvation on a cross or some other such pagan injustice (Deuteronomy 25:1-3; Exodus 21:20 is also relevant to how corporal punishment should never be used to as the means of killing someone), it is not as important that the method is stoning as it is that the capital sinner is executed. All the same, the Bible does mention specific methods of execution, despite many of its capital punishment commands saying nothing beyond that the offender simply must be put to death (as in Exodus 21:12-17 and Leviticus 20:9-13). One might hear people casually say that the Israelites kill murderers by stoning, but does the Bible teach this?
They probably are just repeating hearsay they assumed to be true, and if they were to read the Bible without making assumptions, they would find that it never actually says to stone people for, say, murder. Very often it leaves the means of death completely unaddressed. However, it does teach this, albeit not by its explicit wording. Mosaic Law does clarify enough at times to establish things from which it follows logically that the just punishment for murder, like for many other capital sins, is stoning in particular. Here are two passages that, while not addressing general murder, respectively deal with something very much adjacent to it or with a narrow type of what is murder (immoral killing) on Biblical philosophy:
Exodus 21:28-29--"'If a bull gores a man or woman to death, the bull is to be stoned to death, and its meat must not be eaten. But the owner of the bull will not be held responsible. If, however, the bull has had the habit of goring and the owner has been warned but has not kept it penned up and it kills a man or woman, the bull is to be stoned to death, and its owner also is to be put to death.'"
Leviticus 20:1-2--"The Lord said to Moses, 'Say to the Israelites: "Any Israelite or any foreigner residing in Israel who sacrifices any of his children to Molek is to be put to death. The members of the community are to stone him."'"
If the bull that kills a human (Exodus 21:28-32) is to be stoned to death, and if the owner had warning from others or had personally observed it attacking someone non-lethally yet did not confine it, the animal's owner should also be put to death. The fact that stoning is prescribed for the bull that kills a person and the negligent owner is to share in its fate would require that stoning, unless otherwise specified, is what the person also deserves. The Bible says nothing to illuminate an alternate means of execution, and so stoning would be the default for those who negligently allow a man or woman to die, with malicious, direct murder being the same broad category of sin, but worse.
Leviticus 20:1-5, though, does pertain to intentionally killing someone, and it makes it as clear as language can that stoning is justice here. No sons or daughters were to be killed as part of religious worship (Deuteronomy 12:29-31, 18:9-13). In fact, no person is permitted to kill a human (Exodus 20:13) outside of criminal justice as dictated by God (Exodus 22:18-20, for instance) or in a way that follows by logical necessity from what is prescribed (such as how if the negligent owner of a bovine that kills someone must die, this must be true of other animal owners), or in self-defense (Exodus 22:2-3), or certain cases of warfare, such as battles conducted for the sake of justice or the defense of a community from an unjust power. The murderous sacrifice of children in the name of Molech/Molek or some other deity deserves stoning and thus by extension so would broader murder.
It is plain, also, that stoning is the commonly specified execution method, when it is indeed specified. A sample of the verses that call for stoning in other cases is below:
Leviticus 20:27--"'"A man or woman who is a medium or spiritist among you must be put to death. You are to stone them; their blood will be on their own heads."'"
Deuteronomy 17:2-5--"If a man or woman living among you in one of the town the Lord gives you is found doing evil in the eyes of the Lord your God in violation of his covenant, and contrary to my command has worshipped other gods, bowing down to them or to the sun or the moon or the stars in the sky, and this has been brought to your attention, then you must investigate it thoroughly. If it is true and it has been proved that this detestable thing has been done in Israel, take the man or woman who has done this evil deed to your city gate and stone that person to death."
Deuteronomy 21:18-21--"If someone has a stubborn and rebellious son who does not obey his father and mother and will not listen to them when they discipline him, his father and mother shall take hold of him and bring him to the elders at the gate of his town. They shall say to the elders, 'This son of ours is stubborn and rebellious. He will not obey us. He is a glutton and a drunkard.' Then all the men of his town are to stone him to death. You must purge the evil from among you . . ."
Deuteronomy 21:18-21 prescribes stoning for a very particular kind of disobedience to one's parents--where the child is actively stupid or sinful, the parents have corrected them, and they still persist in their delusional egoism. This is not about children unwilling to submit to a parent's arbitrary, non-obligatory whims (Deuteronomy 4:2, 12:32), which could not possibly deserve to be submitted to. It follows from this, the just punishment being stoning for this exact offense against one's parents on the Christian worldview (not the evangelical/antinomian heresy which contains numerous logical contradictions and Biblical errors), that physically attacking (Exodus 21:15) and verbally cursing (Exodus 21:17, Leviticus 20:9) one's father or mother also deserve stoning, despite how Exodus and Leviticus only say to put such a person to death.
Only marrying someone and their parent at once (Leviticus 20:14) and prostitution (Leviticus 21:9) are prescribed death by burning; to clarify, Leviticus 21:9 might specifically speak of a Levitical priest's daughter deserving burning for prostitution, but men and women are equal (Genesis 1:26-27, 5:1-2), and offenses against them (Exodus 21:15, 17, 20-21, 26-32, and so on) or by them (Leviticus 20:15-16, Numbers 5:5-8, Deuteronomy 13:6-10, and so on) are thus equally severe. Both genders have the same rights and obligations as humans (Exodus 20:12, Leviticus 13:29-39, Numbers 6:1-21, Deuteronomy 512-15, 15:12-17, and so on, including many aforementioned verses). Deuteronomy 23:17-18 explicitly emphasizes how a male and female prostitute are equally guilty, in accordance with what logic necessitates if prostitution is immoral, and logic is true independent of all other things, Christianity included. Leviticus does not say that a priest's son or anyone who is not a priest's child should not receive the same punishment for the same act, and, indeed, Deuteronomy 23 and other parts of the Bible would contradict this, and more importantly the necessary truths of logic themselves.
However, going back to Biblical execution methods as a whole and not their obvious (in light of reason and literal Biblical statements) gender egalitarian application, stoning is certainly the norm. Not only is stoning the general defaut method of execution, but two particular capital sins, of of which is a subcategory of murder, are assigned stoning. Likewise, the inverse is true: although the Torah never directly says that the execution method for all murder must be stoning and instead prescribes it for a subcategory of murder and a sin adjacent to unlawful killing, it also plainly mentions stoning as the regular punishment. A great many sins do not deserve execution (Exodus 21:18-19, 22, 26-27, 22:1, 4, Leviticus 6:1-5, and so on), yes. The sins that do deserve death do not deserve anything more prolonged than stoning, which could easily render someone unconscious and thus unable to experience the remaining blows. This is nowhere near as torturous as even an extended or life sentence in prison, and that is without the physical and sexual abuse that can occur or even be encouraged by emotionalists and hypocrites in this context.
As harsh as it might seem, it would be apparent to any rationalistic person that the typical modern style of punishment (in America, at least) is in no way less severe, not that lesser severity would automatically make it justice, than the system of restitution, flogging, and capital punishment articulated in Mosaic Law--a system that is universally just and therefore required if Christianity is true (Matthew 5:17-19, 1 Timothy 1:8-11, Hebrews 2:2, and more). Stoning is the Biblical default form of execution, which is for men and women and for foreigners and the native-born (Leviticus 24:10-16, 22, and so on), to be initiated by those who witnessed the capital sin (Deuteronomy 13:6-8, 17:6-7). It is not to be paired with corporal punishment like flogging, nor is it to be conjoined with any forms of secondary punishment or humiliation, which in exceeding what is prescribed as justice would by necessity be unjust, and would also contradict the obligation to not degrade people even when punishing them in accordance with God's commands (Deuteronomy 25:3). Stoning is ultimately the almost-exclusive form of Biblical capital punishment, which only very specific sins deserve and which could not become unjust or optional simply because of the death of Christ.