The 57-year-old star died yesterday with family members by his side.
Grey said: "Patrick was a rare and beautiful combination of raw masculinity and amazing grace.
"Gorgeous and strong, he was a real cowboy with a tender heart. He was fearless and insisted on always doing his own stunts, so it was not surprising to me that the war he waged on his cancer was so courageous and dignified."
Demi Moore, Swayze's co-star in the huge movie hit Ghost, said his "light will shine forever".
In the film, Moore played Molly Jensen, the widow of murdered Sam Wheat, played by Swayze. She said: "Patrick you are loved by so many and your light will forever shine in all of our lives
And in the words of Sam to Molly, 'It's amazing, Molly. The love inside, you take it with you.'
"I love and will miss you Patrick."
Swayze shot to fame in 1987 with his performance as dance instructor Johnny Castle in Dirty Dancing, a coming-of-age story set in a holiday resort in upstate New York.
The quirky 1990 romance Ghost cemented his status as a screen heart-throb. He played a murdered man trying to communicate with his fiance through a medium played by Whoopi Goldberg.
Goldberg said: "Patrick was a really good man, a funny man and one to whom I owe much that I can't ever repay. I believe in Ghost's message, so he'll always be near."
Swayze kept on working even after it was disclosed that he had a particularly deadly form of cancer.
He starred in The Beast, a TV drama series about the FBI, and said he and his wife Lisa Niemi were working on a memoir.
In addition to the rave reviews for Dirty Dancing and Ghost, his performance in surf thriller Point Break was well received, and he was nominated for a Golden Globe for his 1995 role as a drag queen in To Wong Foo, Thanks For Everything, Julie Newmar.
In 2006, he made his debut on the British stage, playing Nathan Detroit in Guys and Dolls in the West End opposite Claire Sweeney.
Fellow actors were shocked to hear the news of Swayze's death as they arrived at the premiere of Partir in Toronto yesterday.
The film's star, British actress Kristin Scott Thomas, looked visibly shaken as she passed reporters after learning the news.
Actor Rob Lowe said he had "lost a brother".
He added: "Patrick lived a thousand lifetimes in one lifetime. He was an expert dancer, he wrote hit songs, he starred in hit movies, he was an amazing horseman. But the thing I will remember him most for was his amazing love affair with his wife Lisa."
Lowe added: "He played my brother twice, in The Outsiders when I was 17, and then in Youngblood."
Lowe was at the Elgin Theatre in Toronto for the premiere of The Invention of Lying.