Hmm. Well, in the 5-4 situation, this is always superimportant - I mean, there's got to be some question as to whether Justice Ginsburg will complete another presidential term - 82 is a good innings, as they say in cricket, and 87 is a pretty tough target to aim for, which is one reason why it's vital to the Democrats to replace Scalia now. A Republican victory in November makes it not unlikely that RBG will be replaced by a conservative.
If there's concrete evidence that the justice is not able to do their job, and will not become able to do so with the passage of time. - say, a relative obtains a power of attorney for a justice on grounds of incapacity, due to suffering from Alzheimers, or a traumatic brain injury from which a recovery is not expected - then there is significant pressure for impeachment.
At that point, the President's faction can beginning to push for impeachment on the grounds that to fail to do so is "letting down the American people" or some similar sound bite, implying moral failings and causing reputational damage to the opposition.
The party that doesn't control the White House needs to weigh up what cases are before the court, and how long there is until the next Presidential election and what their chances are there. If there are 40 months to go and the court hasn't got a precedent-setting abortion case, for instance, then giving in now and being statesmanlike about it will help the Senate Judiciary members to look good when it comes time to hit the campaign trail again. If there are 9 months to go, there are a bunch of critical cases before the bench, and the next election is in the bag, then stonewall like a fury, leak that the Justice's condition isn't that bad, spin that it's for life not until it's inconvenient, all that jazz - but keep the justice on the bench even if they have to be nailed there.
If the court is heavily skewed 7-2 or something due to a rash of deaths in a single administration, then there's less to be gained from being stubborn.
If the reason for impeachment is less... morally acceptable... say, busted cooking up meth in the basement or something... the reputational damage for failing to impeach increases. I have a vague recollection that if nobody impeaches a justice, they remain a justice even if they are serving time (and so find it difficult to attend hearings) which might bring a certain immediacy to the argument.