It's not up to him. The way impeachment is designed, it works in one direction. It is Congress vs. the president. One chamber has the power to trigger the trial (that is the House), and the Senate decides whether the charges merit the only remedy available, removal from office (and optional bar from office).
The Constitution does not say the Senate has to conduct a fair trial, that is implied of course, but not expressed in the constitution. In fact, the constitution does not say the Senate must conduct a trial, although it would really be dumb to not.
The senate also doesn't have to allow the president to bring in the kitchen sink. If the charges are weak (as these are,) the senate could easily credibly acquit on a preliminary basis. Like a 12(b)(6) motion in a civil trial (even the allegations don't make a case), the Senate could shut down the articles without getting opposing facts and arguments into the record. Just take the House presentation, and conclude "that's not enough, president doesn't even have to defend himself."
The constitution does not have a mirror image of impeachment - a way for the executive to remove defective legislators or judges. The legislators police themselves, and are only answerable to the electorate.
That is utter BS.