I am sorry you are feeling so embarrassed. I probably would feel the same way, but it still stinks.

One thing that no one has said here, so perhaps you have not thought about it, is that he may be married, in a committed relationship, gay, just out of a relationship and wanting to not date for a while,.... There are quite a few reasons that he may not be interested in dating you while not hating you.

The most likely thing is that he will continue to be polite and helpful and will not bring your letter up in a way that shames you, if at all. That would be the professional thing to do.

The questions I think I would reflect on are why do I follow the impulse? Does following the impulse lead to good outcomes or bad? Is the the feeling of relief of having spoken your mind worth the shame you are also feeling?

Be kind to yourself. You were not horrible-you just acted in a way that makes you feel bad.