Why doesn’t the Republican majority in the United States Senate reintroduce their heathcare legislation?

Loss of Political Will

As many other answers have noted the loss of the House in 2018 made passing the legislation to repeal or even repeal/replace Obamacare in part or whole nearly an impossibility. Any effort to do so would be DOA in the House. That said there is perhaps another reason it has taken a backseat. As described in this NBC News article just prior to the 2018 election many voters were concerned with Healthcare, citing it as their highest priority issue. At the time many Republican representatives in tough races began to move away from hard-line repeal, even if they had previously run on the issue. This is because the electorate soured on that as a solution, particularly when it came to the removal of Preexisting Condition clauses. This article notes that the initial polling on the issue held true though the exit polls and this article further expands on that.

Given that Republicans will be looking to beat back a push to regain the Senate, Mitch McConnell seemingly has learned the lesson described in this article from fivethirtyeight where he will not push legislation in which he has little to gain and much to lose. They have instead focused on judicial appointments, also a keystone of the Republican strategy and one that they have full control over.

While previous answers have correctly pointed out that such a bill would never pass the House, it worth noting that a better line of attack on the ACA by the GOP would be for the law to be ruled unconstitutional. This would require anyone trying to reintroduce a healthcare bill to restart most of the work from scratch.

The ACA bill has already been stripped from the penalty of the individual mandate. This may render all of the ACA unconstitutional.

In fact, as of late June 2020, that is the official line of the DOJ. The case is scheduled to be heard in March 2021 by the Supreme Court of the United States.

A less informed, and more cynical answer than the previous posts: The republicans have no real plan, they are just against anything a democrat is for, and at this point have painted themselves into a corner by promising a narrow set of their base that they will repeal and replace a popular program. They also know that actively appearing to work against healthcare during a healthcare crisis would make them look even worse to most people.

As other answers have noted, any repeal of the ACA is strongly opposed by the Democrats, who control the House, so such a bill has no chance of passing.

Of course, that has not stopped the Republicans from introducing and passing bills through the Senate that they know will be DOA in the House before now, in order to make a statement.

But right now, the world and especially the USA is in the grip of a pandemic. Introducing a bill that strips people of healthcare in the middle of such a pandemic would be incredibly tone-deaf and psychopathic, even for Republicans. Hence they won't do such a thing.

With Democrat control of the House such a bill would have no chance of becoming law. Yehuda points out that bills are sometimes introduced for show, even if they have no chance of becoming law, to demonstrate a party's commitment to making something happen, or so that opponents are forced to vote for or against something that can be used against them later. Why are they not doing this?

The main reason is that the promise to the electorate made by President Trump was to "repeal Obamacare and replace it with something better". The bill that failed in the Senate did repeal Obamacare, but did not replace it with anything (better or not). That was one main reason it failed. There was no replacement because Republicans could not agree on what a 'better" replacement would look like. Many thought that there should be no replacement, but this would deprive many potential Republican voters of affordable healthcare, and remove popular reforms such as being able to get insurance for pre-existing conditions. Democrats are widely associated with the ACA, and so are unlikely to suffer by being forced to vote to keep it.

The likely effect of such a vote (even if not passed) would be to expose divisions in the Republican party over what healthcare should look like, to give opponents ammunition as Republicans voted to remove popular policies, and to draw attention to the fact that Republicans, while promising to "replace Obamacare with something better" are actually voting to simply remove it.

There are aspects of this bill that modify and restrict tax credits. If I'm not mistaken, this would make it subject to the Origination Clause of the Constitution (Article I, Section 7(1)):

All Bills for raising Revenue shall originate in the House of Representatives; but the Senate may propose or concur with Amendments as on other Bills.