Translation:Let's go to the hospital immediately.
Just think about how "immediately" is used in English. In some contexts it means to drop everything, for example "let us go to the hospital immediately" after an accident. In others it leaves some more time, like "we should get working on that immediately" which might referr to any time from right now to next week.
It's the same with すぐ
The urgency implies the need for immediacy in going to the hospital ie. going to the hospital cannot wait - you need to go right now! There's no implication that the speaker and listener/s would go to the nearest hospital although that would certainly be logical and practical. One might presume from the urgency that is implied in sugu that someone has been seriously injured or is seriously ill - hence the urgency.
It didn't like "Let's go right to a hospital." Would it accept "Let's go to a hospital right away"? (I'd personally prefer "Let's get to ..." but daren't try it.) It did accept "right away" the next time around. Next time, "get." Sure enough, they no likee "Let's get to ..."
I think you'll find that in saying "let's go right to a hospital" that that means more "let's go straight/directly to a hospital" ie. without any deviations. Where is すぐ refers to the immediacy of the need to get to the hospital - not that you might get easily distracted from your path - and is best conveyed with "right now/away" or "immediately".
Also, if it were an emergency as we might logically deduce from the urgency of this sentence I imagine you would have already decided which hospital you were going to, in which case "A hospital" is too general - you would instead say "the hospital" indicating that you have a particular hospital in mind already.