The Gunners endured a disappointing weekend after Arsenal lost 3-0 to Premier League leaders Manchester City.
Arsenal produced a very poor performance at Wembley as the Gunners were comprehensively outplayed.
The north London side will switch their focus back to the Premier League and the Europa League in the remaining months of the season.
Arsenal are in sixth place and 10 points adrift of fourth-placed Tottenham following a loss to Manchester City in north London.
The Gunners missed out on a Champions League spot last term following a fifth-placed finish behind Liverpool FC.
Arsenal signed Manchester United midfielder Henrikh Mkhitaryan and Borussia Dortmund striker Pierre-Emerick Aubameyang in the January transfer window.
Former Manchester United defender Neville has ruled out Arsenal getting involved in the Champions League spot.
“There are a lot of big games to come. The race for top four is so important,” Neville told Sky Sports podcast.
“What I would say about Arsenal is that I don’t see them challenging for the top four this season, I think they’re done.
“It’s Tottenham, Liverpool, United and Chelsea for those other three spots. We always said that for Arsenal the top four was the saving grace, but they’ve not got that anymore. In the last few years it’s been a cup, but the cup has gone again here.”
Arsenal will take on Serie A giants AC Milan in the Europa League last 16 after a 4-2 aggregate win over Swedish side Ostersunds last week.
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