Earlier quoted context omitted.
He absolutely should it is his job.
Since you clearly don’t get the context, the Republicans made this a giant issue when Obama had an open Supreme Court seat towards the end of his term. Let’s call a spade a spade: the republicans have no regard for the constitution or laws of the land if they benefit from ignoring it. They will now ignore the precedent they set and place a new Supreme Court justice.
Ultimately, a President with a friendly Senate will always fill the seat. An unfriendly Senate is going to delay the heck out of it. Any rhetoric around that is pretty much just noise. They will send someone through as they've published a couple of lists and have been filling the lower courts pretty quickly. It is not like people weren't aware Justice Ginsburg was very sick.
Frankly, given the President, I would bet it will occur within the next week or two. I do wonder what the polling will be like on this issue. I can actually see a couple of scenarios where Democrats would want to run on a filled seat.