The wife can be corrected for her sin of fornication not only by being put away, but also by words and blows.
(Summa Theologica: Supplement: Q. 62, Art. 2)
St. Thomas had the bad luck to precede Axl Rose in history, so we can’t know whether he would’ve enjoyed a song like Used to Love Her. Aquinas, for his part, reckoned that an offended husband lacked the prerogative to kill his unfaithful wife (the only exception being if he were to catch her in flagrante delicto, lose his senses completely, and fly into a murderous rage on the spot. The criminal codes of the time neither condoned nor punished such an action). If he wanted to kill her after the fact, the husband would have to have her tried, convicted, and executed by the secular authorities:
There is no doubt that a husband, moved by zeal for justice and not by vindictive anger or hatred can, without sin, bring a criminal accusation of adultery upon his wife before a secular court, and demand that she receive capital punishment as appointed by the law; just as it is lawful to accuse a person of murder or any other crime.
(Summa Theologica: Supplement: Q. 60, Art. 1)
The husband’s own capacity as head of the household did not extend to that of executioner. Why? Thomas reasoned that it was because a dead woman can not make amends:
There are two kinds of community: the household, such as a family; and the civil community, such as a city or kingdom. He who presides over a community of the first kind, can inflict only corrective punishment, which does not extend beyond the limits of amendment, and these are exceeded by the punishment of death. Wherefore the husband who exercises this kind of control over his wife may not kill her, but he may accuse or chastise her in some other way.
(Ibid.)