For western Europe, marriage and divorce were governed by the canon law of the Latin Catholic Church, or at least that’s what the church argued. Secular authorities did try to claim authority over marriage but certainly by the 12th century it was recognized that the church was the authority in spiritual matters, including marriage. But canon law wasn’t really fully developed until the 12th and 13th centuries, so there was some disagreement about how to interpret it and what could and couldn’t be done. Therefore the answer is…maybe! But probably not.

The most influential compilation of canon law was by Gratian, who took all the canon law he could find in various different sources and attempted to make them coherent (the “concordance of discordant canons”, or more simply, the Decretum). He listed 8 grounds for divorce, but adultery was not one of them. So around 1140 when the Decretum was published, it would be the opinion of the church that no, adultery could not lead to divorce.

There were other opinions about how interpret canon law though. According to another 12th century canon lawyer, Rufinus of Bologna, there were also 8 grounds for divorce, but “fornication with a third party” was one of them - with some restrictions. Divorce could occur only if the marriage hadn’t been consummated yet. If the marriage had been properly consummated, then divorce was impossible.

“The decretists unanimously agreed that while the Church had the power to separate a couple when one party could prove that the other had committed adultery, separation created no right of remarriage for either party so long as the original partner lived.” (Brundage, Law, Sex, and Christian Society, pg. 294)

A couple therefore did not have to live together or act as a married couple, but separation isn’t divorce - and if they were unwilling to reconcile, they still had to remain legally married. They could live with other people but then they’d both be adulterous fornicators according to the church, and they certainly couldn’t legally marry anyone else, since that would be bigamy.

There was also a question of who could initiate a divorce, even under the limited circumstances where divorce might have been possible. A man could accuse his wife of adultery based on rumours or suspicion, but a woman had to have actual evidence that her husband was an adulterer. Some canonists argued that a man could divorce his wife for adultery, but a women couldn’t divorce her husband, no matter how strong her evidence. Other canonists argued that the couple could divorce (not just separate) but they couldn’t remarry at all. Some argued that divorce should never be allowed at all, for adultery or any other reason, because it was such an important “spiritual good”.

The lawyers also wondered about something that must happened frequently enough: if a divorce was granted, could the adulterous party (man or woman) marry the person they were committing adultery with? Some said no, some said yes. What if the original spouse had died? In that case they could remarry…as long as the adulterous spouse had not murdered the other spouse simply so they could remarry.

Canon law was further refined by the popes in the 1230s into the Corpus Iuris Canonici, which became the official law of the church until the 20th century. In the Corpus, the answer was generally a firm no - adultery could not lead to divorce, except in the limited circumstances where the marriage had not yet been consummated. Divorce was allowed for other reasons (if one spouse wanted to join a religious order, the husband was impotent, one spouse was mentally incompetent, among other things), and annulment was always an option but that’s a different question entirely.

Maybe the most famous case of annulment was Eleanor of Aquitaine and Louis VII of France. Their marriage was ultimately annulled because of consanguinity - they were too closely related according to the rules of the church. Consanguinity was usually ignored (because basically everyone was too closely related!), unless people were looking for an excuse not to be married anymore - in this case, Eleanor had failed to produce a male heir. But in the process of annulling the marriage, Louis looked at all the possible options, and apparently he considered accusing Eleanor of adultery. There were rumours that she had had an adulterous, incestuous affair with her uncle, so maybe that’s what he had in mind. But everyone agreed that proving adultery would be impossible, so they went with consanguinity instead.

Sources:

James A. Brundage, Law, Sex, and Christian Society in Medieval Europe (University of Chicago Press, 1990)

James A. Brundage, “The canon law of divorce in the mid-twelfth century: Louis VII c. Eleanor of Aquitaine”, in Eleanor of Aquitaine: Lord and Lady, ed. Bonnie Wheeler, John Carmi Parsons (Palgrave Macmillan, 2003)

Depending on location and timing, you wouldn't necessarily even need them to cheat on you.

The Old English penitential Scriftboc does indeed does indeed condone divorce in cases of spousal infidelity, but simply not wanting to be married any more was also justification enough. West Saxon law allowed divorce if it was mutually agreed by both parties. If the agreement was not mutual, one party would be forced to grant a divorce if the other wanted to go into a religious community.