Trailing 2-0 from the first leg, United only managed a 1-1 draw against their fierce rivals at Old Trafford, which saw Liverpool go through to the quarter-finals.
However, Van Gaal praised his side's performance and highlighted Sunday's Manchester derby as a key fixture in United's bid to finish the Premier League season in the top four and qualify for the Champions League, per the Press Association (via MailOnline):
I am not angry, I am not frustrated. I was very proud of my players. They gave everything and I was very pleased the fans recognised that. They applauded after the match despite us being out—that was remarkable I think. We have created a lot of chances; we scored a penalty, [chances for Jesse] Lingard, [Juan] Mata, [Marouane] Fellaini, [Anthony] Martial, and then you concede a goal in the last seconds.
That is what we did not do in Liverpool, they scored an away goal and that makes a difference how they feel in the dressing room at half-time. In the second half we tried to raise ourselves out of disappointment but you know, the fans know, everyone knows we have to score three goals and that was more difficult.
There is always pressure. Expectations are very high so pressure for these players is always very high. We can cope with that but we were not lucky today. I hope City is the catalyst. We have to beat City and we have a chance to still qualify [for the Champions League].
Anthony Martial gave United hope of a dramatic comeback when he netted a penalty after half an hour, but Philippe Coutinho's delicate finish on the stroke of half-time left the Red Devils needing three more goals to go through, per BT Sport Football:
BT Sport Football @btsportfootball
.@Phil_Coutinho's beautiful finish sends @LFC on their way to the #UEL quarters... https://t.co/ISHkDXNlhu2016-3-18 07:15:06
The hosts never even got close, and goalkeeper David De Gea was once again United's star player, an indication as to how well-beaten the Red Devils were in the tie, per the Times' Oliver Kay:
Oliver Kay @OliverKayTimes
De Gea far and away #MUFC best player as they're soundly beaten by an in-transition #LFC over two legs. What a story that tells.2016-3-17 21:55:34
Van Gaal's future at Old Trafford is seemingly hanging by a thread.
Winning the Europa League looked United's best chance of qualifying for next season's Champions League, but their exit means they must try to finish in the top four.
It will not be easy, as Van Gaal's men are four points back from fourth-placed Manchester City with nine games of the season remaining.
And United have shown no indication that they have the ability to string the necessary results together to force their way into the top four.
After a run of four wins on the bounce, United have reverted to type and have not picked up a victory in their last four matches.
The insipid performances that put Van Gaal under pressure in the first place have returned, and it would be hugely surprising if he were still in the top job at Old Trafford next season.
Football writer Sam Pilger gave a damning assessment of United's season under the Dutchman:
Sam Pilger @sampilger
Van Gaal does the Treble: Dreadful in the Premier League, Champions League and now the Europa League #mufc2016-3-17 21:51:17
Jose Mourinho is viewed by many as Van Gaal's likely successor, and it was even previously reported that there is an agreement in place that would see the Portuguese installed in the summer, per Samuel Luckhurst in the Manchester Evening News.
Bleacher Report's Dean Jones believes it would be a good appointment:
Undoubtedly a derby win against City on Sunday would improve the mood at United, and it would also move them to within just a point of the top four—unless West Ham United beat Chelsea away on Saturday.
However, the limp way in which United exited the Europa League and their consistently poor performances this season have put Van Gaal on shaky ground.