The shadow Treasury minister called for the poll as part of a package of sweeping constitutional reforms, including the introduction of proportional representation.
Speaking at a campaign event in south London, Mr Lewis said: "Why not have a referendum in this country on the future of the royal family?
"We are a democracy. I'd rather see us as citizens rather than subjects in the 21st Century.."
He added: "Let's look at what a modern state looks like and what the role of the royal family would be."
The shadow minister also said that Labour must "modernise or die".
Speaking in Brixton, Mr Lewis said: "We can't have more of the same. The Labour Party needs to modernise or it will die.
"I'm fed up with the top-down style of politics, where real debate and discussion in our party is stifled because of sectarianism and tribalism. We can't grow as a party if we're afraid of having difficult discussions.
"I'm standing because I see a party in crisis and democracy in crisis, and unless we start addressing some fundamental issues, a few tweaks of policy here, or a slight change of leader there, aren't going to bring the real change that this country urgently needs."
Mr Lewis and shadow foreign secretary Emily Thornberry face a race against time to remain in the battle for the top job.The latest figures from the Labour Party showed Mr Lewis and Ms Thornberry are the only two in the six-candidate field who have failed to reach the threshold of 22 nominations.
Mr Lewis has four signatories - 18 shy of the target figure - and Ms Thornberry has nine.
Shadow business secretary Rebecca Long-Bailey, Lisa Nandy and Jess Phillips all secured the 22 backers required to continue in the competition on Thursday.
As of Friday, Ms Long-Bailey currently has 26 supporters, Wigan MP Ms Nandy has 24 and Ms Phillips has 22.
Early front-runner Sir Keir Starmer, the shadow Brexit secretary, was the first to cross the line earlier this week.
He stretched his lead over the rest of the pack, with the party confirming that, as of Friday, he has 63 backers.