Stubborn as always, Professor Higgins doesn't rise when everyone else does in order to attend Doolittle's wedding.  (Doolittle, of course, is Eliza's father.)  From Act IV, here is what the text says on the matter: 

LIZA. Quite. Is the Professor coming?
MRS. HIGGINS. Certainly not. He can't behave himself in church. He makes remarks out loud all the time on the clergyman's pronunciation.

The reason on the surface, then, is that Higgins behaves just as badly in church as he does at the race track.  It is not appropriate, of course, to correct the priest's pronunciation.  (Ha!)  Higgins always has to interject his own opinion whever he is, whether it is appropriate and socially acceptable or not.  In my opnion, this is why Mrs. Higgins is so exasperated by her son all the time.  Mrs. Higgins is a prime example of an English socialite and is incredibly embarrassed by her son's inability to be the same.  Therefore, he remains a continual embarrassment to her.

Below the surface, however, the reason is that Professor Higgins doesn't actually belong at Mr. Doolittle's wedding.  He has no relation to the bride, the groom, . . . or the groom's daughter (Eliza) as anything but a business associate or former "boss," so to speak.  Professor Higgins stands aloof from everyone and, as such, wouldn't be welcome or even enjoy this blessed event.  So as always, Professor Higgins stays home stewing about the exasperating events of the past few months with chagrin.