Mr Gaffney said he would be taking part in equality and diversity training following his speech at a Labour Students Burns Supper in Edinburgh.
The Coatbridge, Chryston and Bellshill MP made a joke about Scotland’s national poet not being “bent” in what was supposed to be a jocular piece of doggerel.
He also used a derogatory word when describing a Chinese meal.
Mr Gaffney said: “Last week I attended a Labour Students Burns Supper In Edinburgh.
“At that event I used certain language relating to the Chinese and LGBT communities that was wrong and completely inappropriate.
“I want to offer my unreserved apologies for what I said; my remarks were deeply offensive and unacceptable.
“I will be taking part in equality and diversity training at the earliest opportunity.
“I will do everything possible to make amends with both the Chinese community and the LGBT community.”