Graham Stringer, MP for Blackley and Broughton, said letting the people have their say on the issue was the "minimum" response required to the poll in which Labour narrowly beat the Conservatives into third place.
Miliband was urged to back a referendum on the EU, to win back Ukip supporters
He told the BBC: "While these elections were about Europe and there was certainly some protest vote, there were also some people voting against the direction this country is going in Europe.
"Unless we have a policy response to that, which has to be as a minimum to give people a referendum, then we are going to lose votes.
"It is a very unattractive policy to say vote for us but we can't do anything about your major concerns because Europe won't let us. So I think we have to improve our offer on Europe. We can't just keep saying this has been a major wake-up call."
He also criticised the organisation around Miliband and claimed it was "elitist and arrogant" to keep saying the party would simply carry on along the path to greater integration in the EU.
"At the moment I believe we can do much better if we tighten up our organisation around Ed Miliband and we change some of our policies so we have more resonance with the electorate," he continued.
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European and local council elections
"I think it is very elitist and arrogant to carry on saying we will just go the way that Europe is going with ever deeper integration. That isn't what people I represent want. It is not what people in the UK want.
"So all I want to do is improve the Labour party's position for the next general election. We have to have some policy and organisational changes."
Bassetlaw MP John Mann wrote on Twitter that the Labour party would ignore Ukip at its "peril", adding: "Labour drew false comfort before 1992 'just one more push'. It would be foolish to repeat that error."
But shadow business secretary Chuka Umunna said Labour had done well last night.
He told Sky News: "This was a disastrous result for the Tories coming third it looks like, the worst ever result for them.
"The Liberal Democrats wiped out in certain places like in my own borough in Lambeth in London.
"Actually, I think we can be confident, we can be pleased with our progress. It's not in the bag, we can't say that at all, we have to keep working to build on the support we have already won amongst the British people.
"We are approaching the big one next year with humility."
He added on the BBC that he was "not at all" interested in becoming Miliband's successor.
"I backed Ed's leadership campaign before anybody thought he could become leader of the Labour party because I thought he was the one best qualified to become prime minister," he continued.
"That is still the case and any talk of the soap opera nonsense you get around this place is totally self-indulgent. What everybody is focused on in the Labour party is ensuring we have Ed Miliband as our prime minister this time next year so we can deliver a better tomorrow."
With only Northern Ireland left to declare, Ukip had secured 24 MEPs out of the 73 UK seats available, representing an an increase of 11 on 2009.
Labour had won 20, an increase of seven and the Conservatives 19, a loss of seven. The Liberal Democrats were beaten into fifth place by the Greens, returning just one MEP.