Meyer, 26, proposed to Palin, 24, Friday night at a Rascal Flatts concert in Las Vegas, Palin said on her blog
"The lead singer, Gary LeVox, dedicated 'Bless the Broken Road' to us, and then Dakota got down on one knee and proposed!" she wrote. "It's amazing to see what happens when you place everything in life in God's hands."
Friends and well-wishers congratulated the couple on the news. As one person said in a comment on Meyer's Instagram, "This is so 'MERICA that my mind is blown right now."
Meyer received the nation's highest military honor
for braving enemy fire in 2009 to recover 36 American and Afghan troops in Afghanistan. He was the third living recipient -- and the first Marine -- to be awarded the Medal of Honor for actions in Iraq or Afghanistan.
Palin, the oldest daughter of former Alaska Gov. Sarah Palin, has stayed in the public eye since her mother's failed vice presidential bid through appearances on "Dancing with the Stars" and a Lifetime show about raising her son, "Bristol Palin: Life's a Tripp."
She also released a best-selling memoir in 2011, "Not Afraid of Life: My Journey So Far." She was previously engaged to Tripp's father, Levi Johnston.
The couple met last year when Meyer visited Alaska to film "Amazing America," Sarah Palin's show on Sportsman Channel
, Bristol Palin said in her blog post.
"He met Tripp during that time, and I've seen him in a few places where our paths have crossed since. He's visited us in Alaska, and I've visited his wonderful family in Kentucky," she wrote. "He's wonderful with Tripp and I'm so proud to be marrying him."
The couple shared news of their engagement on social media.
"I'm definitely the luckiest guy ever," Meyer said in an Instagram of the pair tagged #shesaidyes.
Bristol Palin shared an Instagram picture of a ring and another of the happy couple smooching.
"Truly the luckiest girl in the world, cannot wait to marry this man!!!!" she said.
The couple started recently began sharing images on social media of themselves together.
Palin posted a photo last week of the couple out in the snow with Tripp. Meyer has also shared a few pictures of him with Sarah Palin.
"Thank you for being the amazing person that you are. I appreciate your leadership and dedication to our country!" he said in a post on her birthday in February.
Sarah Palin also congratulated the couple, noting that Meyer "flew up North" to ask Bristol Palin's father and grandfather for her hand in marriage.
"Our families couldn't be happier for Bristol and Dakota! We're honored to welcome Dakota into our family. He's an American hero and patriot whose service to our country -- like all his fellow Medal of Honor recipients -- has been above and beyond the call of duty; but even more important is he's a good and kind man who loves Bristol and Tripp, and is loved by them," she said on Facebook
"Yes, God working behind the scenes to turn beauty from ashes. Had tough circumstances in their lives not occurred, and had they reacted differently to those circumstances, they'd have never met."