Ramos was suspended after picking up a booking in the first leg against Ajax which sent him over the disciplinary threshold, and he will be banned for one more European game after he suggested he received the caution on purpose.
The Blancos captain watched on from a box as Solari's side capitulated and his replacement at centre-back, Nacho, was sent off in the closing stages.
Speaking after the game, Solari said: "Without talking bad about any player, of course we missed the captain.
"It's still a painful week, the fans have been spectacular, they've been supporting a difficult moment, they have not caught up with us.
"They are three times European champions, today we have been eliminated, it is a very sad moment and it puts in context what they have done before, it is very difficult to win.
"Madrid always comes back and always comes back stronger."
The victory means Ajax will advance to the quarter-final stage, and head coach Erik ten Hag admitted: "We took maximum advantage of Ramos' absence."
Los Blancos are 12 points behind the Catalan giants in the Spanish top tier after losing to their Clasico rivals on Saturday.
The latest loss marks Solari's third in seven days, and his job is now beginning to look untenable with Jose Mourinho, Mauricio Pochettino and Massimiliano Allegri among those tipped to replace him this summer.
"It's really painful, we're sorry for our fans, who were amazing in the way they supported the team in what was a very tough time," he said.
"We're sad and are hurting. We put in the best possible effort but it wasn't enough and we've bowed out. It's a sad night.
"A lot of things went on out there. It was a game in which things frequently changed. There was a bit of everything: enforced substitutions, shots hitting the woodwork, the ball not wanting to go in, their clinicalness, the VAR system… An endless amount of things that you'll have to assess."
And with former key players Gareth Bale, Isco and Marcelo also looking likely to depart Santiago Bernabeu in the near future, a considerable turnaround in the Spanish capital looks imminent.
Chelsea forward Eden Hazard has made no secret of his desire to join Real after saying he dreamed of playing for them as a child.
But Los Blancos will first turn their attention to a clash with Real Valladolid on Sunday.