The Argentina international met head coach Michael Carrick and senior officials on Thursday before releasing a statement acknowledging that the timing and setting of his remarks had been wrong.
“Today I met with the club's leadership and the manager,” Martínez said in a social media post.
“It was a very positive conversation, where we were able to exchange views and speak openly about what happened after Tuesday's game.
“Everyone knows that I am a passionate person and that I live football with great intensity.
“The frustration of the defeat led me to express how I felt at a time and place that perhaps were not the right ones, and for that I want to apologise.”
The controversy began after Martínez questioned whether United had strengthened sufficiently during the summer transfer window. He singled out director of football Jason Wilcox while arguing that the squad needed more investment to compete at the highest level.
The defender also made clear that his frustration was not intended as a criticism of Carrick, whose side have endured a difficult start to the campaign.
According to the BBC, Martínez’s comments have resulted in an internal disciplinary response, with the defender understood to have been fined. United have not publicly confirmed the details of any punishment.
Martínez’s statement also sought to draw a line under the disagreement and reinforce his commitment to the club.
“Our club is at its best when we are all united, and I recognise that my words in that moment did not help with that.
“I know that every single person at Manchester United is doing everything possible to achieve our shared ambition: bringing the success that this club and its supporters deserve.
“With this situation now behind us, we are fully focused on winning together.”
The episode has unfolded during a difficult run for United. They have taken six points from their first six Premier League matches, been eliminated from the Carabao Cup by Brighton and lost to Atlético in Europe.
The club must now decide whether Martínez remains part of its longer-term plans. His contract expires at the end of the season, although United hold an option to extend it by a further year.
His apology resolves the immediate public dispute, but it does not remove the wider questions over United’s recruitment strategy or the future of a defender who has challenged the club’s leadership over its ambitions.