Smith made unsubstantiated claims questioning Eubank's sexuality during a pre-fight press conference on Thursday.
Eubank, son of former world champion Chris Eubank, wore a rainbow armband to Friday's weigh-in.
He wrote on Twitter: "We don't discriminate... we don't alienate. We want boxing (and) sport as a whole, to be all inclusive."
Both fighters could face sanctions from the British Boxing Board of Control after Eubank responded by accusing Smith of infidelity to his wife.
Smith has apologised for his comments.
"It's crossed a line and for people looking at it, it's wrong," said the former WBO light-middleweight champion.
"We shouldn't act like that. It's fight week, we're both starving and making weight. If anyone took it personally, I apologise."
