ABU DHABI—Russia and China are stepping up their efforts to woo Persian Gulf arms buyers, encroaching on a market long dominated by the U.S. and Europe and raising security concerns in Washington.

The market is crucial for the global defense industry, with Saudi Arabia’s $7.7 billion in 2018 deals making it the Mideast’s top arms buyer, and the United Arab Emirates and Qatar also in the world's top 10, according to market-research firm IHS Markit.

Saudi...