If a U. S. citizen marries someone whose citizenship is that of a European country (member of the EU), does the U. S. person automatically become a citizen of the European country of their spouse? Or is there a process similar to that in the U.S., where there must be a petition, etc.?
I am not aware of any European country that “automatically” grants citizenship to spouses - at the very least you’d have to be married for a certain number of years before you’d be eligible, and yes, you would have to fill out an application.
The exact procedure varies by country, though.
In the UK, getting married to a citizen allows to you a work permit, and one year’s residence. If, after one year, you are still married (and residing!) together, you can apply for a further two years. If the two-year period passes without incident, you can then apply for citizenship (at least, that was the law when I was involved in the process), and the application can take up to a year or two.
If, at any time during the process, you are (a) divorced, (b) separated, (c) convicted of a crime, you will lose your right to stay in the UK, or, at least, your next application to whatever stage you’re in will be rejected. Something I learned to my peril when (b) then (a) applied to me. 