Garcia's men took a 1-1 draw into the home leg of the last 16 tie at the Stadio Olimpico but found themselves 3-0 down after 22 minutes as their first-half performance descended into farce.
Mohamed Salah hit the crossbar and post after the interval as Fiorentina tried and failed to add to a 4-1 aggregate triumph amid fury from the home fans who remained in the ground, with the iconic Curva Sud virtually deserted.
Roma face a battle to qualify for next season's UEFA Champions League having won only twice in Serie A since the turn of the year and Garcia struggled to conceal his horror at the manner in which their European campaign crashed to a halt.
"It was an unexpected defeat," he told Mediaset Premium.
"Fiorentina deserve compliments for making the most of our gifts to them, but I am mortified. We had the right desire at the start, but paid a heavy price for individual errors.
"Now we have to salvage the season with second place. If we qualify for the Champions League, it will be an acceptable campaign."
Amid growing discontent from the terraces - Roma last won at the Olimpico in November - Garcia has insisted that his players must show character to get the supporters back on-side.
"There are 11 battles remaining in Serie A. It’s just the staff and the players. I will not let go for a second, we have to lift our heads and get back up.
"This is what I expect from my players. I spoke to them after the game, telling them that anyone with pride and character will follow me. If anyone is weak, then they do not interest me.
"I talk about character because some players seem to be afraid of playing at the Olimpico. Our stadium must get back to being a pleasant venue for us.
"In this moment of difficulty that's the message I want to give to the team."
He added: "The leader must set the example and I remain motivated. The season can still end well. We need more character."
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