(1) needs some punctuation, but it is debatable how best to do this. " is a possibility. However, unless you have to literally transcribe spoken words, the problem can be avoided by writing "I just want to know whether my eye has recovered".
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yellowstarstruckIn speech is it ok to say "I just want to know- has my eye recovered? Thanks, FulvioIn speech, yes.
GPYIn speech, yes.I should clarify that it's not actually wrong in everyday writing. In formal writing I would avoid it though, and use the "whether" alternative that I mentioned.