One objection is that it's somehow unethical or unprecedented to confirm a SCOTUS justice in an election year. I wrote about this recently and won't go over the same ground here.
It's enough to say that it's neither unethical nor unprecedented. Twenty-nine times there has been a SCOTUS vacancy in a presidential election year, and in every case the president made a nomination. In 19 of those cases the presidency and the Senate were in the hands of the same party and 17 of those 19 nominees were confirmed. In the 10 instances when the Senate and presidency were controlled by different parties only 2 of the 10 were confirmed. In fact, if the president's opponents insist that he not exercise his constitutional responsibilities in an election year they're implicitly demanding that he limit himself to a three year term of office rather than four.
This is ironic because the Democrats were adamant in 2016 that the Republicans give Obama nominee Merrick Garland a vote before the election. From The Washington Free Beacon:
Obama, then-vice president Joe Biden, Hillary Clinton, then-Senate minority leader Harry Reid (D., Nev.) and Senate Democrats ripped Republicans, who were in control of the Senate, for their decision not to consider Garland. Sen. Patrick Leahy (D., Vt.) said the Senate had approved judges as late as "September" in election years, and Sen. Bernie Sanders (I., Vt.) said the notion that Obama could not nominate a judge that year was "absurd."Another objection is that President Trump doesn't have time to get a replacement for Justice Ginsburg confirmed before the election or before his term is over, but this concern lacks historical support. There are 42 days until the election. Justice Ginsburg’s confirmation in 1993 took 42 days, John Paul Stevens's confirmation in 1975 took 19 days, and Sandra Day O’Connor was confirmed in 33 days in 1981.
Now the situation has reversed, as Democrats insist President Donald Trump and Senate Republicans not move on the vacancy left by the death of Justice Ruth Bader Ginsburg.
Even if Trump loses the election on November 3rd and the Republicans lose their majority in the Senate he's still the president until January 20th and the Republicans are still in control of the Senate until January 3rd. That gives them plenty of time to confirm a nominee. In fact, every nominee of the last 45 years has been confirmed in less than 110 days.
There's nothing unconstitutional in nominating and confirming a justice this close to an election. Here's what the Constitution says about the president’s power to appoint justices to the Supreme Court courtesy of National Review's Jim Geraghty:
He shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur; and he shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law: but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments.That's the entirety of what the Constitution has to say about the appointment of justices. Geraghty adds this:
It [the Constitution] doesn’t say anything about how close the vacancy is to Election Day. It doesn’t say anything about whether the Senate has to hold hearings about the nominee. It doesn’t say anything about whether the Senate has to vote on that nominee; a refusal to vote on the nomination, as occurred with Merrick Garland, is a de facto rejection. It doesn’t say anything about whether the Senate can vote in a lame duck session.In other words, as long as Mr. Trump has fifty votes in the Senate, which is by no means a sure thing, the nominee he announces later this week will almost certainly be the next Supreme Court justice.
This means President Trump can nominate anyone he likes up until noon on January 20, 2021, if he isn’t reelected. The Senate can choose to hold a vote on that nominee anytime it likes. Or it can choose not to hold a vote on that nominee. If the Democrats win a majority of the seats in the Senate, they take over on January 3, 2021. If the Senate is a 50-50 split and Joe Biden wins the presidency, then Mike Pence breaks ties up until January 20, and then Kamala Harris breaks ties in the afternoon.