Yes, the moderator should call him out because some of Trump's statements are so easy to prove false. He was in favor of the Iraq War, he has said horrible things about women, he has claimed the election is rigged, and the list goes on. A moderator should always call his bluffs.
Yes, the moderator during the presidential debate should call out Donald Trump's, as well as Hillary Clinton's, false claims whenever possible. I believe that the debates should be used by the candidates to address the issues facing the country, and to expand upon their political platforms, not to spread misinformation. The moderator has a responsibility to not only thoroughly question the candidates, but press them when their answers are evasive or inaccurate. Rigorous fact-checking in real time allows the voting public to make informed decisions when selecting their candidates.
The moderator should have the responsibility to call Donald Trump out for making false claims during the debate. The majority of the potential voters watching the debate have no idea whether what he is saying is correct or not. The moderator should let the public know when claims being made are false so voters are more educated about the candidates.
The moderator should rely questions from the crowd/network and control the timing of each candidate. Candidates in any debate will give false claims, but there are fact checkers who release the information online during the debate. If the moderator calls out on candidate on false claims, then both candidates should get called out every mistake they make, which would turn the debate into a discussion of who can list the most facts.