The 21-year-old put on a fake Slovenian accent and pulled a face as she mimicked the future US first lady while hosting the awards show in Los Angeles.
In a handwritten note posted on her Twitter account on Monday, she said: "I removed or changed anything in the script that I felt took the joke too far, and whether or not you choose to see it, what remained was done in good humour and with no bad intent.
"I believe Melania understands show business and the way shows are written and run. I apologise to anyone that I offended and have only the best wishes for our country. "
During her impression, Hadid joked about Slovenian-born Mrs Trump's Republican National Convention speech.
"This is my Melania Trump impression. I've got to get the face right," Hadid said.
Adopting a fake Eastern European accent, she added: "I love my husband, President Barack Obama and our children, Sasha and Malia."
Hadid - who is dating former One Direction singer Zayn Malik - faced a barrage of criticism on social media following her impression.
One Twitter user wrote: "I'm not a Trump fan at all but Gigi Hadid's frankly racist impression of Melania Trump is really unacceptable."
Another tweet said: "What @GiGiHadid is wrong with you? So disrespectful. What if someone did that to Mrs Obama?"