"You will not support this hospital anymore."
Translation:Du wirst dieses Krankenhaus nicht mehr fördern.
duo's translation is fine, in my opinion. to support something (as in 'hold up') = stützen or tragen in german.
'Die Stiftung wird mit öffentlichen Mitteln gefördert.' You support (unterstützt) someone in the kitchen, with a real helping hand. You can support (unterstützen) an idea or movement and hence you support it financially (fördern). Also, this thread hopefully supports (fördern) someone's understanding of the difference between those words.
The Duden says: unterstützen = behilflich sein [be of help, give a hand, assist] and to support someone in a bad material/financial state with money. For the latter use, you have to mention the money in some way [mit Geld, finanziell, materiell, etc.] or it must be clear from the context. 'Our club is broke, we have to support it' - Unser Verein ist pleite, wir müssen ihn unterstützen. The third definition is yours, to support someones ideas, concerns, affairs.
fördern = to help someone (often financially, not always; see my 3rd ex.) with his/her/its development, evolving, advance. You usually don't donate money to a hospital because it's out of money, but to maintain or improve it.