I don’t think I’ve ever heard any specific ones, I’ve heard people come up with specific blessings on the spot but just the usual “¡Felicidades!” or “¡por muchos años!” as a standard formula. The literal meanings are “may you be happy”, “long years (of happiness)” but they’re just the Spanish multi-purpose ways to congratulate someone.
And if someone who just got married is a new couple, the officers involved should’a waited until the booze wore off…
Similar to **Nava’s **example, there were cries of “ad multos annos” after a traditional Catholic Latin mass wedding that I was at earlier this year. It seemed quite appropriate given that the bride and groom had exchanged their vows in Latin too.