In Judaism, if a man walks into his wife having sex with another man, is he permitted to kill both, the other man and his wife, on the spot?
And the punishment for this is strangulation as Rabbeinu Yonah writes in Shaarei Tshuva (3:129-130)
The following sinners are executed by chenek (strangulation): [...] one who cohabits with a married woman.
Indeed the next sentence in Shaarei Tshuva is
Our Sages of blessed memory said: From the days the Temple was destroyed, even though the four death penalties of beit din have ceased, the punishment of the four death penalties has not ceased [i.e., the sinner dies in a manner similar to the penalty prescribed for him] [...] one is liable to chenek either drowns in a river or dies in a choking disease.
ABSOLUTELY NOT. The bystander is only allowed to use lethal force in halacha to PREVENT a murder or a rape; and even then, only if there is no non-lethal option to prevent it.
This is all spelled out quite clearly in Maimonides' Laws of the Murderer and Preservation of Life, Chapter 1:
Deut. 22:26 says we don't blame a married woman who was raped: "just like if someone was murdered, [you wouldn't blame the victim]; the woman cried out, but no one was there to save her." Jewish tradition reads that as saying *but if anyone was there to prevent the rape, there would have been no constraints vis-a-vis Jewish law from doing so", i.e. a bystander could use force -- up to lethal force, if necessary -- to prevent it. And because the verse is comparing rape to murder, the same applies to preventing a murder. [1:10]
"Prevention" means just that -- before a rape or murder has occurred. Witnesses and bystanders are not allowed to kill someone who has just committed a murder; that person needs to be tried by the courts, as Num. 35:12 states the murderer shall not die until he stands before the community for sentencing. The same applies to any capital crime -- no death penalty until tried by court. [1:5] The same applies to rape -- lethal force may only be used to prevent it from occurring in the first place. [1:12]
Lethal force is only authorized if non-lethal force won't do it. If a bystander could cut off the attacker's hand, break his foot, or blind him to prevent the rape/murder, do that instead. [1:7] If the bystander could actually have prevented the rape/murder non-lethally but said "eh why bother" and instead kills the attacker, such a bystander is considered a murderer in God's eyes! However the courts would not punish them. [1:13]
To elaborate mbloch's point, and answer your comments:
#1. AFAIK, there are only a couple of cases when one is allowed to kill a Jew without being killed himself (by the court):
In the case of a Rodef (pursuer), one is allowed to kill him without bringing the pursuer to justice.
In the case of one who was sentenced to death by the court and got away.
In all other cases, a transgressor(s) must be brought to justice. In other words, if you kill a person that violated any capital sin without a trial, you'll be sentenced to death (Of course, it requires witnesses and all the accepted procedures, but you ARE a murderer).
#2. Assuming the wife is Jewish, a man has no special rights toward his wife's body, and legally for him, she's just another person, not as his slave. The husband cannot beat her or injure her or force himself upon her or kill her (see below).
#3. Halachicly-wise it might be permitted to judge and execute a non-Jew
In your case, if you're interested in killing the person you should provide two independent witnesses that would testify what you saw. (THere's a dispute in Sanhedrin 5-6, I think, if a husband can testify against his wife - פלגא דיבורא?)