US citizen marries a non-citizen do they become a US citizen?

No, marriage to US citizen does not automatically allow a non-citizen to become a permanent resident or citizen of the US.

Marriage does entitle the non-citizen to apply for US permanent residency, provided they can show they will be living in the US. For example, if a US citizen married a German citizen, and they continue to live in Germany, the german citizen cannot apply for a US permanent resident visa. If, at some time later, the couple decide to move back to the US, the German citizen can at that time apply for a permanent residency visa. Such marriage-based permanent resident visas are almost always granted once the couple shows proper residency intent (and, shows that the marriage is not a sham or fraud one).

Any non-citizen holding a US permanent residency visa can apply for US citizenship via the nationalization process. The key requirements of the citizenship process are:

1. Be a permanent resident for at least 5 years (or 3 if married to a US citizen)
2. Reside continuously (except for short-duration excursions) in the US for that period of time
3. Have a clean criminal record, or obtain a specific waiver from an Immigration judge
4. Attend proper citizenship classes