No, [House} Democrats1 could not have stalled the confirmation by using impeachment.
The exception, "unless otherwise ordered by the Senate", means the Senate could, "after the trial shall commence" delay any impeachment trial(s) until after confirmation.
1 The House impeaches, the Senate tries the impeachment.
The Senate can choose to disregard the rules and if a majority supports that decision there is nothing that can be done to override it. The U.S. Senate did so at one other point, at least (the quorum required for a committee vote), in this particular nomination.