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The chief executive of Robinhood Markets Inc. waded in Thursday to the ongoing debate surrounding off-exchange trading, saying that financial regulators should allow public stock exchanges to quote securities prices in hundredths of a penny in order “to level the playing field.”

In a statement on Robinhood’s blog, Vlad Tenev said the online brokerage supports amending Rule 612, which was adopted by the Securities and Exchange Commission in 2005 and generally prevents exchanges from quoting prices in increments of less than a penny.

Meanwhile, off-exchange platforms—where individual investors’ trades, including those of Robinhood customers, are often executed—are able to buy and sell stocks at sub-penny prices.

“In a nutshell, exchanges cannot fairly compete with off-exchange market makers in executing our customers’ orders,” Mr. Tenev said.

The debate about where and how trades are routed has recently garnered heightened attention, in part because of the rush of individual investors into the market this year.

Market observers have become increasingly focused on a practice called payment for order flow, from which online brokerages such as Robinhood make money by routing customers’ orders to electronic-trading firms. Though the practice is commonplace in the brokerage industry, Robinhood in particular has found itself at the center of the payment-for-order-flow debate this year.

Online brokerages and electronic-trading firms say the practice benefits smaller investors by executing their trades at slightly better prices than they would get elsewhere. Yet the concept has also drawn scrutiny because of concerns that diverting trades away from exchanges makes markets less transparent. Nearly half of U.S. trading volume now takes place off public exchanges, a share that has increased in recent years.

Newly confirmed SEC Chairman Gary Gensler outlined this month a wide examination of market structure, and suggested that nonprofessional investors might get better prices if more trading were done on exchanges.

“The question is whether our equity markets are as efficient as they could be, in light of the technological changes and recent developments,” Mr. Gensler said at an industry conference this month. He also said the SEC’s review would examine the minimum price increments that exchanges are allowed to offer.

Under Rule 612, exchanges can only display stock prices in fractions of a cent if a security is trading at less than $1 per share. Mr. Tenev said Thursday that if sub-penny limitations are removed, “we could see…perhaps more retail order flow executed on lit markets.”