This is not a direct relationship. Health insurance is not a limited-supply product, on which demand can drive up prices.
In fact, it's the opposite. Any one company can write coverage for every single person in the world. There is no "supply" problem at all -- and better, the more people an insurance company covers, the lower their risk and therefore the less they have to charge. So driving up demand will actually DECREASE prices.
But indirectly, the actual HEALTH CARE is limited. If twice as many people try to go to the doctor, they will not be able to find enough doctors, so doctors would be able to charge more -- until we get more doctors through medical training.
However, we could help solve that problem if we loosened restrictions on who could be primary care physicians. Doctors are overtrained for that role, which is why under HMO rules a lot of that work is done by nurse practitioners. You bump up to a doctor if the problem's solution isn't obvious.
If we relax restrictions, some people can be gatekeepers after a 4-year college degree, driving up supply and lowering costs since those gatekeepers won't have 10 years of expensive medical school to pay for.
TO do this, you need tort reform, so people can't sue when the gatekeeper makes a mistake.
To people who consider any interference with the health care market an affront to capitalism, consider this: many people who have pre-existing conditions can't enter the job market and those who have something go wrong and have insurance can't ever leave their job, making the labor force much more static than it should be. I am fortunate (not B.S. “fortunate”, but God-given fortunate) not to have these problems, but I know people who do - and their skills are underutilized or not utilized at all in their present positions.