Remember, Aragorn was not king yet. He had not claimed his kingship when Frodo was stabbed. So, not being king, he coud not heal like that. Tolkien can be very technical in writing and wording. I think he did this, to emphasise how kingly Aragorn was, when he claimed it.
But, I don't think Aragorn said that in the books anyways.
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Do Not Touch
-Willie
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