The phone, which uses the Android operating system, received generally positive reviews but analysts said it may not be an "iPhone killer" for the South Korean manufacturer that leads the global mobile phone market.
Apple and Samsung are fighting patent battles in more than half a dozen countries. Each company accuses the other of infringing on patented technology in smartphones or tablets.
But Apple last week backed away from a bid for a temporary restraining order to block the import of the Galaxy S III into the United States, even though the patent case is still pending in a California federal court.
The newest Galaxy is being offered by several US carriers, and Samsung hopes to win over some loyal iPhone owners with the smartphone's features.
USA Today's Edward Baig called the Galaxy S III "a top-notch Android handset."
"Most human beings will like the Galaxy S III, as I did. After all, it's a phone designed for us," he wrote.
Hayley Tsukayama of The Washington Post called the Samsung device a strong contender with an impressive design slightly slimmer than the iPhone.
"Is it an iPhone killer? Let's call it a worthy contender," she wrote.
Wired magazine's Nathan Olivarez-Giles called it "Samsung's most ambitious smartphone yet," but said it fell short of the iPhone and failed to outperform the HTC One X among Android handsets.
"As a whole package, the S III simply doesn't feel like a finished product. It could use more polish, more thought, and a more elegant user experience," he said.
The Galaxy S III has so far been launched in more than two dozen countries, mainly in Europe and the Middle East. It will be available in 145 nations by July.
The third version of the Galaxy S series offers face-recognition technology and improved voice-activated controls as well as a more powerful processor that lets users watch video and write emails simultaneously.
It can detect eye movements and override the phone's automatic shutdown if the user is looking at the screen.
Samsung is now pinning its hopes on the S III to further erode its rivals' market share before the expected new version of Apple's iPhone 5 this year.
More competition will be coming from new Windows-based phones, and Google is expected to expand its offerings though its newly acquired Motorola Mobility.
Samsung still leads the global smartphone market. According to ABI Research, the Korean firm sold 43 million smartphones in the first quarter to 35 million for Apple.
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