In his statement the president, in part, offered this: "I simply ask Republicans in the Senate to give him a fair hearing and then an up or down vote. If you don't, then it will not only be an abdication of the Senate's constitutional duty, . . ."
As a supposed Constitutional "scholar", clearly the president must have realized his statement was in error - the US Constitution specifies no such "duty".
The Appointments Clause, i.e., Article II, Section 2, Clause 2, states:
Article I, Section 5, Clause 2 states:
The Senate Majority Leader has indicated his opposition to considering any Supreme Court Justice nominee until after this year's election cycle has ended.
In part the Senate Majority Leader cited the unofficial "Biden Rule" as a precedent.
What is the "Biden Rule"? It is named after the current Vice President, Joe Biden, who in 1992, as a Senator, gave a speech on the appropriateness of nominating a Supreme Court Justice during a presidential election year. At the bottom of this offering is a link to the actual video of Biden's remarks.
Almost immediately after the Senate Majority Leader's statement stories started showing up decrying that people were taking Biden's statement out of context. Even Biden, himself issued a statement to this effect.
Next is the actual video of Joe Biden from 1992.
But, first, this link offers additional opinions on this subject in general.