If "provide for the common defense" can be used to justify spending as much on the military as the next 10 countries combined then perhaps "promote the general Welfare" might be considered to include keeping the citizens of the country healthy.
You're quoting the preamble, which simply explains why they wrote the Constitution. The actual powers of the federal government are vested in Article I, Section 8. Specifically, items 12, 13, and 14:
12: To raise and support Armies, but no Appropriation of Money to that Use shall be for a longer Term than two Years;
13: To provide and maintain a Navy;
14: To make Rules for the Government and Regulation of the land and naval Forces;
There is no specific power granted for "promote the general welfare". Item 1 may come close, saying that "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", but the fact that the founders didn't start building hospitals means that they never meant it that way. It's something the states can do, much in the same way that states establish university systems.