Baldwin, 55, announced the decision in a statement, saying that his comments were "offensive and unacceptable." His talk show "Up Late with Alec Baldwin" began last month.
"I did not intend to hurt or offend anyone with my choice of words, but clearly I have - and for that I am deeply sorry," Baldwin said in the statement, adding that he was trying to protect his wife and daughter in New York.
Earlier this week, the actor tearfully testified at the trial of a woman who was sentenced to 210 days in jail after being found guilty of stalking and harassing him and his wife, Hilaria.
In a video posted on celebrity website TMZ.com on Thursday, Baldwin confronts a photographer filming him in a New York street and calls him a homophobic slur that drew the ire of gay rights group GLAAD, which has previously defended the actor when he was accused of homophobic language.
"Mr. Baldwin can't lend his support for equality on paper, while degrading gay people in practice," GLAAD spokesman Rich Ferraro said in a statement. "It's clearly time he listens to the calls from so many LGBT people and allies to end this pattern of anti-gay slurs."
It was not known if the photographer at whom Baldwin directed his comments was gay.
Baldwin, who has had many confrontations with photographers and reporters approaching him in public, also came under fire in June for homophobic tweets aimed at a reporter for Britain's Daily Mail newspaper.