AFAIK (and in my experience), it's not standard practice to not grant any time off in the first six months of employment. However, at my work, no one is granted PAID vacation time off during the first six months because it is the probationary period so an employee is not eligible to use paid vacation time.
Also, if an employee states at the time of hiring that there is already a commitment coming up that requires time off, there typically isn't an issue.
People are getting really stuck on the fact he was hired to work every Sunday. I got hired to work every Monday-Friday; my job is a Monday-Friday job. I don't get the option to state, "I will now work Sunday-Thursday because I need Fridays off. This doesn't mean I can never have off one of my regular workdays, but it sure does mean I can't have one of those days off on a regular basis. The guy was hired to work Sundays and was told this as it's their busiest day (I worked at a retail pharmacy 20 years ago and was told Mondays were our busiest day so everyone worked on Mondays, which was true as all of us were regularly scheduled on Mondays). When he informed his employer he would not work Sundays moving forward, he decided his own fate. He also was told he'd be working Christmas then turned around after two months of employment and stated he wanted Christmas off. Two months, he'd been working there two months and wanted Christmas off over the other chefs who had been working there longer. Sometimes, no matter how busy or how shorthanded you know you will be, it's best to let go the deadweight (and, yes, not so much Christmas but the statement about no longer working their busiest day of the week tells me he's deadweight). I'd be interested to see how well his next job pans out.