can an ordained minister marry themselves?

It depends on the state law for the jurisdiction where the wedding is taking place. Some states specifically forbid it, others do not specify and some explicitly recognize it to accommodate religious groups like the Quakers that have this as part of their tradition.

Can Ministers Marry

In almost all states, the answer is no and it has nothing to do with church leaders. It's the law, and the person doing the "officiating" serves a legal function, partly to ascertain they are both getting married under free will. You can't have the same signature on a marriage certificate for the groom and the officiant for the same reason the bride and groom can't be their own witnesses.

It depends on the local laws.

In Colorado and Pennsylvania, a couple can solemnize their own marriage without an officiant regardless of whether one of them is a minister or not.

In most other states, the minister would have to be a separate person from the couple.

Sometimes, yes! Check with your state and local laws.

No. No one can marry themselves. It must be done by a separate person who can ascertain that the couple is there of their own free will.

no