I agree, and that's one of the reasons that I think private health provision on a national level is bad thing.
I think your view of how a private health system works/would work is very skewed.
There are a few things at play that you don't seem to take into account (Using the US as my only base of knowledge here):
1) There are 122 medical schools in the US and they all have their medical graduates take the
Hippocratic Oath or
Declaration of Geneva, so every doctor trained in the system has stated a moral obligation to treat every patient that it is in their power to treat. The Declaration of Geneva even specifically states sexual orientation.
2) Very few doctors are their own boss. Nearly every private practice is either owned by a larger company or multiple doctors who partnered in the business. Unless that one very specific ownership group has a discriminatory policy then doctors would be expected to work within a contract of moral and ethical rules and would not be able to maintain their job and discriminate at the same time. Even if they did allow a doctor to discriminate, they have a staff of multiple doctors.
3) While a libertarian world would be unlikely to require a licensure process, licensure would still exist. Much like any other job you hire someone for, you want the guy that is heavily credentialed. I'm currently doing a lot of work on my house. We have had electricians and HVAC guys coming in and giving us quotes. There is no law requiring a license to perform these jobs, but we only took quotes from fully licensed professionals who were able to show their credentials. Licensure boards would be likely to create their own set of rules for a doctor to maintain their license, if necessary.
4) Also like any other private industry, in today's world it would be harder for a discriminatory doctor to pass under the radar. This is the world of Yelp, Urban Spoon, and Trip Advisor. Or, as is more fitting,
this is the world of HealthGrades.com, RateMD.com, and Vitals.com. Just like I checked Yelp when looking at contractors I check Health Grades when looking for a new doctor.
In a purely private system could homosexuals (or others) be discriminated against? Yes. Could the doctor get away with it and easily be able to continue practicing? Most likely not.
Should a doctor be forced to take on a patient he doesn't want to? No.