Mr Leonard said he is "taking the flak" for an "incredibly bad" result, where the party lost its two MEPs and slid from second to fifth place, with 9.3% of the overall vote in Scotland.
He played down suggestions of tumult within the senior ranks, following the resignation of his close ally, Neil Findlay, from the front bench with a call for the party to end infighting and a "toxic" culture of leaking and briefing.
He said: "I'm not going to resign as leader of the Labour Party.
"We've had a constructive meeting of the group, the first chance we have had to talk about the result of the European Parliamentary election.
"There is a determination in the group to find a constructive way forward.
"We recognise that 9% is an incredibly bad result but we need to come together to build a way from that."
He added: "I am taking the flak, and I am prepared to take the flak, for what was a poor result.
"Sometimes when your back's against the wall, people come together.
"We need to rebuild the Labour vote but we will do that by being united."
He repeated his assertion there should be a second EU referendum on a "credible" deal to avoid the threat of a no-deal Brexit.
One of the Labour MEPs who lost their seat was David Martin, the UK's longest serving MEP, having spent 35 years in Brussels.
Following the result declaration on Monday, he said the party "lost not because of lack of effort but lack of a clear message".
He said Labour leader Jeremy Corbyn had become "stuck in limbo" on the question of the UK's EU membership.