Can you get a marriage record sealed from the public?

Sometimes. In US jurisprudence, there is a public interest in a fully open record. However, if you have a very compelling reason that convinces a judge s/he should keep the case sealed, sometimes they will. Usually, though, only parts of a record are sealed, particularly if the issues deal with sensitive subjects, like the privacy of children, the frail, issues regarding third parties not involved in the litigation, or the handicapped, etc.

No. marriage records are public records.