I have an intern at a fitness club who is a very hard worker. On paper, he meets all of the qualifications for a job opening except for his ability to communicate well to customers because he has a heavy Asian accent. How do I tell him that his communication level disqualifies him from the job without sounding like the company is discriminating based on ethnic origin?

You prove it. You set up 5 minutes of scenarios that each job applicant must go through, talking to you as though you are a customer. For instance, a new customer wants to know how many pounds of weights he can lift to start. Maybe two or 3 other questions too.
I would have 1 or 2 other people listen to answers too, so that you aren't the only judge.
It's somewhat subjective. I can usually understand phone reps speaking heavily accented English when other people can't. If he's a hard worker, I'd probably hire him anyway. He might be a good source of new customers.

Is the communication listed, has he been warned about this in his current position, has he been talked to before about it, Or has all of his current reviews been good ?

Has there been any customer complaints, and were they docujmented.

Next why not offer him some english as second language or other training to make him a better employee