"I was able to talk to him backstage after the pageant," the actress and model, 26, tells PEOPLE. "He approached me, he said sorry, I accepted his apology. He didn't have to apologize, but I'm grateful that he did, and I think that speaks very highly of his character."
Wurtzbach says she admires Harvey for his ability to own up to his error.
"I thought he was such a gentleman to do that," she says. "I think it takes a lot for you to admit you made a mistake and correct it right away, especially on live television. And I understand that it's live television – you can make mistakes. It's human, it's okay."
The pageant queen says she has also spoken to Gutiérrez since the competition.
"During Christmas Day, which also was her birthday, I wished her a happy birthday and merry Christmas, and said, 'I wish you're doing okay,' " said Wurtzbach. "She replied back to me, so I think everything's okay."
While her crowning moment was less than ideal, Wurtzbach says she was just happy that she won the coveted title.
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"I didn't really mind because I was so excited that I won," she says. "It is my childhood dream. Of course, I was also concerned for Ari. I could only imagine what she must be feeling at that time. I couldn't really celebrate just yet because I knew that the girl beside me was broken."
Because of the mix-up, Wurtzbach never got to take her first walk as Miss Universe, but she will be making up for it at a homecoming ceremony that will be thrown for her in the Philippines later this month.
"They're going to invite everyone to come with their flags, and then they'll give me my first walk," she says. "We'll get to relive that moment, and I'll be able to do my first walk in my home country."
As the first Miss Philippines to take home the crown in 42 years, she's become a major celebrity in her native country.
"I'm still on the front page of the newspaper and I'm still on the news every day," she says. "Everything I post on social media, they make a headline out of it!"
As Miss Universe, Wurtzbach hopes to bring attention to a number of causes – including HIV/AIDS awareness, disaster relief and cyber-bullying – and is looking forward to traveling the world.
"I hope to meet a lot of people from different walks of life from different parts of the world," she says. "I'm looking forward to just enjoying my reign as Miss Universe."