In 1866 Congress passed a law providing that no vacancies on the Supreme Court would be filled until the total number had dropped to 7--this was done to prevent Andrew Johnson from nominating anyone to fill vacancies. Two Southern Unionists on the Court died while he was President but he was not able to name any justices. Roger Taney died in 1864 and Lincoln picked a new chief justice, but the next new justice was appointed in 1870, chosen by Grant.
I'm sure Schumer would love to pull something similar to prevent Trump from making any more nominations, but to pass a law over Trump's veto he would need 2/3 in each house.
To increase the size of the scotus would require a dem potus, dem majorities in both houses of congress, and the nuclear option in the senate (unless dems already had 60 votes there).