They can investigate too, if they so choose.
But I was answering your question about Trump's course.
I'm saying that they should require Trump to disclose the evidence he has that convinced him this is a fact, not merely a suspicion. Until that evidence is received, they really have nothing to go off of. Again, the accusation was very specific. It would be a "witch hunt" if they merely were investigating any possible surveillance efforts, legal or not.
With the Russia/Trump issue, there is evidence that Russian officials attempted to illegally influence the US election in favor of Trump. And, no one claimed that it was a "fact" that Trump's administration colluded with the Russians in any specific way. And, we all know that Trump's own surrogates/appointees cast suspicion on themselves by providing false statements regarding meetings with Russian officials.
With Trump's allegations of Obama, Trump stated that it was a "fact", that the wiretaps were ordered by Obama himself, and that they were illegal (not via judicial approval). And, there has been no evidence that his allegation stands muster. Evidence would have to show that Obama ordered illegal wiretaps on the Trump Campaign at Trump Tower in NYC.