Stopping a judge
The only ways to stop the nomination are for the Senate to vote against him or the president to withdraw the nomination.
The only way to remove a federal judge is an impeachment. That requires a majority vote in the House of Representatives, a trial in the Senate, and at least two thirds of the Senate to vote against the judge.
So as a practical matter, only the president or the Senate can stop a judicial nomination, and only Congress can remove a judge afterwards. So the Senate is key here. A majority now or a two thirds majority of the Senate in the future would be required to remove a judge.
What's "not qualified" mean?
On another hand, I am not sure what the Bar Assn. committee decision fully entails. I take that it does not refer to the Bar Exam, considering that this individual has actually practiced law (albeit so far only for 3 years).
They don't say. Apparently they interview former associates of the judicial candidate in coming to their rating. They give three different ratings: very qualified; qualified; not qualified.
Republicans apparently think that the problem is that the candidates are pro-life (in favor of allowing laws restricting abortions). As such, they reject the depiction of the candidates as "not qualified" by the American Bar Association. It's unclear what is ultimately going to happen. Traditionally Republicans have felt that the ABA was biased against their nominees. The may simply stop participating in the ABA ratings or start their own vetting organization. They already stopped ABA vetting prior to nomination.
This contrasts with President Barack Obama who had fourteen not qualified candidates that he then did not nominate (see the previous link for more info). He had the ABA vet his candidates before he made the official nomination. So he could quietly pick another candidate without announcing the failure of the first one. Trump's not vetting before the nomination nor supporting the ABA vetting at all.
Stopping this nomination
To get back to your question, to stop the nomination, you either need to convince a majority of the Senate to vote against the nominee or you need to convince the president to withdraw it. At this point, no other organization matters at all.
It seems unlikely that President Trump will withdraw the nomination. He is more likely to view this as a challenge from the ABA. His natural response to challenges is to fight them. That leaves the Senate.
Since Harry Reid changed the Senate rules to not allow a filibuster of lower court judicial candidates (district and circuit/appeals), you would have to convince at least fifty-one Senators to vote against him (assuming Vice President Mike Pence would vote for him in a tiebreak). That means that at least three Republican Senators would have to vote against him. Both his home state Senators have declared him highly qualified (in the Politico article), so it's not clear that this will be viewed as a Trump thing. Senator Ben Sasse (of Talley's home state of Nebraska) is a noted Trump critic but supports Talley.
As a practical matter, they would probably have to find three Republican Senators to vote against him and hold every Democratic Senator.
TL;DR: You would have to convince enough members of the Senate to vote against the nomination.