Belgian side Club Brugge stunned Atalanta 3-1 in the second leg of their playoff tie to dump the Italian side out with a 5-2 aggregate win and reach the last 16.
Lookman, Atalanta's hero last season when they won the Europa League, pulled back one goal for the Italian side when they were 3-0 down.
But the Nigerian, who has scored 15 times this season, had a penalty saved, which Gasperini said Lookman was not supposed to take with offensive midfielder Charles De Ketelaere and striker Mateo Retegui on the pitch.
In a post on Instagram, Lookman said Gasperini's comments singling him out were "deeply disrespectful" and hurtful.
Gasperini brushed off the quarrel while addressing a press conference on Saturday, pointing out similarities to an incident in Udinese's 1-0 Serie A win over Lecce on Friday where Lorenzo Lucca scored a penalty despite not being the designated taker.
"It had an extraordinary impact, everything that came afterwards... mine was not meant to be an offensive sentence," Gasperini said.
"Yesterday in Udinese-Lecce we had what could have been a drama, luckily there were guys who let it go. I would have liked a strong player like Lookman to have made a gesture towards De Ketelaere saying 'come on keep the ball, put it in'.
"He can become a penalty taker, that would be an additional goal-scoring ability... I always speak in front of the whole team, it has never touched the club. He felt offended, I didn't want to offend anyone."
Atalanta, third in Serie A with 51 points from 25 matches, travel to relegation-threatened Empoli on Sunday.
"Empoli come from a streak of difficult results, all the games I've seen they've performed very well," Gasperini said.
"They are a good team, with a good quality of play, with an excellent coach, who are playing for salvation. I remember the first game, I expect a balanced, difficult match. We only have one goal, we have to do our best."