AFAIK he simply fills the role of the President of the Senate, which is basically to sit there and look pretty while the real work happens down below. He might have to rule on something, but any ruling of the president can (AFAIK) be overturned on the spot by the senators. (That’s why presiding over the Senate is normally a job for junior senators. It’s boring and powerless.)
Executive summary: He called the Senate to order. He told the chaplain to read a prayer. He recognized Senators to speak. When a Senator made a motion, he asked if there was unanimous consent and ruled whether there was. When a Senator raised a point of order, he asked the parliamentarian what the right answer was and so ruled. He ordered the clerk to call the roll when required and ordered the clerk to stop calling the roll when a motion to stop was made and consented to. Then heard a motion to take the Senate into private session and ruled that the “ayes” have it. A Senator asked for a 10-minute break, the CJ ruled that he heard no objection, and he ordered a 10-minute break followed by a private session.
END OF SUMMARY.
The presiding officer in the Senate actually has very little official power. He can’t even initiate a quorum call, let alone declare sua sponte that the whole proceeding is out of order and dismiss the Senate.
I suppose a Senator on the floor could move to have the entire proceeding dismissed with unanimous consent and the CJ could rule that he hears no objection while Senators are jumping up on their desks screaming “objection.” waving their hands and stomping their feet. But if he pulled a stunt like that, the next order of business for the House would be to impeach the CJ and bring another impeachment proceeding against the President.