Mr Eastwood said there were far-right protestors outside Taoiseach Simon Harris’ family home on Thursday night, describing it as “utterly disgraceful”.
He added: “We’ve also seen this week the new first ever black Mayor of Derry facing an enormous deluge of abuse from around the world, right up to and including very serious death threats.
“And this kind of stuff is allowed to happen actually because politicians don’t mind their language and I think all of us have to be very careful about that, whether it’s on this island or whether it’s in the government in London.
“It’s important I think that all of us stand against that level of racism so that this is a different kind of place and we will not be led into the gutter by far-right activists, whether they’re coming from America, Dublin or Derry online.”
The SDLP has been criticised after two other candidates, Shauna Cusack and Jason Barr, said a selection convention was ‘cancelled’.
The pair claimed they were “removed” from the selection process and that an online interview was held instead.
They have both quit the party and will now sit in the council as independents.
Mr Eastwood has disputed that they were excluded, saying there is a “new process” to decide on the Mayor which was “unanimously supported” by the party membership.
However, he admitted that there are “lessons to be learned” about how the party communicated how that process is going to work.
Said the Foyle MP: “But I do think it’s important to say Lilian Barr was the standout candidate. She is a phenomenal candidate.
“Of course, it’s a hugely historic moment that Derry will now have the first ever black mayor, someone who came from Kenya years ago, got herself elected with a big mandate onto Derry city council.“And beyond all of that is an extremely qualified and talented person to represent the SDLP but, actually, more importantly, to represent Derry and to show that we’re a city, a city of civil rights that won’t put up with that kind of racism, will stand up for a more pluralist society.
“That’s the values of the SDLP, it’s the values that are represented by the people of Derry, that comes all the way from the civil rights movement and we’re very proud in this moment to have Lilian representing us and Lilian representing the people of Derry.”