The interim head coach was speaking to the media for the first time since part-owner Sir Jim Ratcliffe's controversial interview when he claimed the United Kingdom had been "colonised by immigrants".
The comments have been widely condemned, including by Prime Minister Sir Keir Starmer and all United's leading supporter groups.
Ratcliffe did offer a qualified apology, while United released their own statement highlighting the inclusivity initiatives they are part of.
"I'm proud of what the club stands for and has done for so long," said Carrick. "I'm really aware of that.
"Sir Jim made his statement and the club made a statement on the back of it. It is not my place to add to that.
"What I can say is that I have been around this club many years.
"Equality and diversity and respect for each other is something we look to carry through every day.
"I have travelled the world and know what this club means to an awful lot of people.
"I am fully aware of the responsibility, and we try to carry that out every day. I am proud of what the club stands for and has done for so long."
While Ratcliffe was a known supporter of Brexit and his views around immigration were well known, the language he used in his latest assessment has been questioned.
In particular, it has been argued that while he already had a high profile due to his status as one of Britain's richest people and a hugely successful businessman, his association with a global brand like United means he should be more careful in what he says.
Carrick, who has been installed as head coach for the remainder of the season following Ruben Amorim's dismissal, wore a distinctive green Unite for Access badge.
The organisation aims to raise awareness and "drive equality for a more inclusive matchday experience for disabled sports fans".
"I think part of being at this club is that we understand what it's like globally and the responsibilities we have within this," Carrick added.
"I can only speak from my personal experiences of being here for a long time with various managers, with various staff.
"The supporters have largely probably stayed the same but we have gathered some new ones over the years, from all different backgrounds. It's something we're fully aware of and I am."
