An index of popular songs and tunes by musical phrase does exist. Indeed it is quite famous. It is not online as far as I know; it is a book published in the middle of the last century. Unfortunately, I can't remember the name of the author or of the book but I remember finding it in the library in the '60s. I did do a search online a few years back and found information about him. He became a professional consultant on plagiarism. In fact, I think there is an article about him on Wikipedia. He is deceased and so his book is of value for older songs only. I will try to find info about him again and if I do, I will do a follow-up post.
 
Okay, I did a search and found that Google has applied for a patent for just such a search engine it is US 6678680 B1. Read about it at: www-dot-google-dot-com/patents/US6678680 ,substituting a dot for -dot- (this site doesn't allow URLs)
 
Guess what? A Swiss database does exist where you can search by note sequence. It appears to have classical themes only, unfortunately, but here it is: www-dot-rism-ch-dot-org/pages/database
It could be useful if someone accuses you of plagiarism. If you can find a similar phrase in the Public Domain classical literature, you will probably destroy any case they thought they had. It could also be useful to consult before you accuse someone else of plagiarising YOU.