No. He also wore shoes, slippers, sneakers and went barefoot.
He only wore them when making Western movies. If you saw " The Quiet Man " , you would see that he dressed for the period in Ireland and his shoes were like every one else. He wore the same shoes as other people when not acting for a Western. So, No he didn't always wear cowboy boots.
Well, he dawned his dancing shoes, too, black patent leather ones and danced the jitterbug in this movie:
No. He did own a ranch in Arizona, so I'm guessing he wore them often.