Originally Posted by Xurtio

The most obvious way is to have a better understanding of the physiological mechanisms so you could measure it in the brain and compare brains of lots of people .

Psychologists already have a stress-diathesis model (two different people exposed to the same environmental stimulus - one might trigger a mental illness due to biological differences) which is part of the more general "biopsychosocial" model - that is, acknowledgement that influence in mental illness outcome depends on a combination of social, biological, and psychological factors.

These factors are separated by monozygotic twin studies, and those might inform you somewhat to the degree which illnesses (or homosexuality) may be social vs. biological in general, but to further correlate them would require some careful case by case statistics and intervention studies (but it would obviously be unethical to try to induce mental illness in people) to establish causation. You could start a homosexual support group and see if correlation between homosexuality and mental illness goes down in 20 years, but then you'd also be ignoring any biological changes and assuming them fixed.