Tom Watson said his party could not “sit on the fence” about the biggest issue to face the UK for a generation.
But ex-UKIP leader Mr Farage said a new referendum would be “a total insult” to five million Labour Leave voters.
The UK has been given an extension to the Brexit process until 31 October.
This means the UK is likely to hold European Parliament elections on 23 May.
Mr Farage launched his new Brexit Party last week and said it had a list of 70 candidates to fight the May elections.
Mr Watson warned that Labour would not defeat Mr Farage “by being mealy-mouthed and sounding as if we half agree with him”.
“We won’t beat him unless we can inspire the millions crying out for a different direction,” he added.
He said a “confirmatory” referendum and “final say” on any deal was “the very least” voters deserved, now they knew more about what Brexit would mean.
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