Yes, you can certainly file a lawsuit. One of the major flaws of our judicial system is that anybody can file a lawsuit about anything without risk to their own financial interests. Unlike the legal system in every other Western country, the United States does not employ a "loser's pay" philosophy.
While my tenure in law school was a mere ten days, I know enough to tell you that you are highly unlikely to win such a case. By the time your case came to trial, your problem would have corrected itself.
You would be better off by expending your energies to have the situation corrected. What you developed was eyelid ptosis. This usually takes place about 48 hours after injections, but can occur as late as two weeks. Most often the defect lasts 2-6 weeks, depending on the dosage hitting the levator palpebrae muscle. Eyedrops can correct the problem. Iopidine ( cute name since IOP stands for Intraocular pressure; Apraclonidine) Naphcon-A (naphazoline) or Mydfrin (phenylephrine) can all be used..not at the same time.
These drops stimulate Muller's muscle and this will elevate the droopy eyelid. Usually, 2 drops , 2-3 times a day until the ptosis corrects itself on its own.
Incidentally, during the infancy of Botox injections, we were told that eyelid ptosis was due to the fact that a certain percentage of the population lacked a pseudomembrane which protected the eyelid. However, now it seems that the reason for eyelid ptosis was poor injection technique. Studies at the time, with expert injectors, showed an eyelid ptosis rate of 5.4%. Now, the rate is considered to be roughly 3%, which some studies as low as 1%.
This accents the fact that 100's of patients a week develop this problem (among millions of Botox injections world-wide)...its lack of rarity would not bode well in court.