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I am sorry that your cat is not feeling well. Yes, cats can get viruses, but most of the time we will not see vomiting as sign of it, but more often respiratory signs. The causes of vomiting can be many and varied and could be something as simple as an allergy to his food or problem with his stomach not emptying fast enough to more serious things like intestinal obstruction or metabolic disorders from an internal organ disorder.
Even though he is no longer vomiting, if this lethargy continues, it would be a good idea to have him examined. There may be signs of underlying problems that your vet may pick up on an exam and my even recommend further testing. If he would let you, you could take his temperature rectally, but even if there was no fever present, the decrease in activity alone would be enough to warrant an exam.
I hope this is helpful and please let me know if you have any further questions.