This is also the basis for federal involvement in education, environmental law, and social security, among other things. You may not agree with it, but you are not going to win any lawsuits arguing against all of these things. This is well-established law. In fact, we've had government-run health care for decades (Medicare, Medicaid, and the Department of Veterans' Affairs) and the Supreme Court has never ruled it illegal.The Congress shall have power To lay and collect taxes, duties, imposts and excises, to pay the debts and provide for the common defence and general welfare of the United States
At any rate, it certainly has nothing to do with the First Amendment as the OP claimed.