Google didn't say that. It said prostrate meant that. Now, the original English sense of 'prostrate' is 'face-down on the ground' but it calls back to Latin prostratus which is actually laid forth or out in the manner of a slain enemy or KOed boxer. Similarly, the other senses of 'prostrate' mean 'leveled', 'defeated', 'powerless'.
No one is going to bat an eye if you use prostrate in your sentence but if you want to avoid some potential ambiguity you just need to say lying prostrate: lying in a prostrate fashion, which calls out the helpless, immobile, powerless senses of the word.
The word that will actually make people think he's facing the wrong way in his bed would be prone. As @MarkHubbard pointed out above, its opposite is supine although that's relatively less well known.