Is it allowed to go to a hospital named after an idol with an idol on display in the lobby?
By giving them money, you support them to continue, and they are doing so called "good works" in the name of their idol, which is a known common way of worshipping their idol.
Thank you for your help.
It's easy to think "everything is idols, I can't go anywhere or do anything!" If you study tractate Avodah Zarah (especially with the commentaries that tried to square it with actual Jewish practice in Medieval France/Germany), you'll see there's a lot more nuance than that.
For example, the only bathhouse in the neighborhood had a statue of the goddess Aphrodite on it, yet Chief Rabbi Gamliel used the bathhouse. He explained to a surprised non-Jew: "Honestly, you guys didn't say let's build a bathhouse to glorify Aphrodite, you said let's build a bathhouse, and put up an Aphrodite to decorate it."
I can't speak for Eastern faiths or practices; but in the English-speaking world, let's assume we're talking about a typical Catholic hospital: "Saint Joseph's", "Good Samaritan", you get the idea. Yes there will be icons and/or crucifixes. On a good day I would bet that easily half of their patients, and staff, are not practicing Catholics.
In pages 192--193 of Mesoras Moshe Volume II, Rabbi Moshe Feinstein's secretary mentions taking him to Saint Vincent's Hospital for a checkup, and Rabbi Feinstein saying it was okay to call it "Saint Vincent's." He also entered a hospital (same or different, not clear) with a big crucifix in the main lobby. He said it was better if reasonably possible to enter a different way, but ultimately it was allowed -- nobody in their right mind would think you're entering a church.
It seems it was just a matter-of-fact that the hospital was the hospital.
As for your supporting their practice ... sooner of later some of my money is going to wind up in the hands of someone who will do something I don't like. The Talmud makes the crucial differentiation between enabling and simply being involved. If I sell a pagan a half-pound of frankincense and am the only shop in town, I'm enabling him to go burn it to his deity. If I sell a hundred pounds of frankincense, wholesale, to a shop, I'm sufficiently removed from their customers. It's thus very hard to argue that me paying my hospital bill is THE thing that's enabling non-monotheistic worship.
As for "doing kindness" being their way of worshiping ... going to be very hard to avoid that. I think we'd also fall back on the involvement-vs.-enabling argument above.
