The former Florida governor made the comment as part of a discussion on strengthening families and lifting people out of poverty during the Faith and Family Presidential Forum at Bob Jones University.
"This is a place where I think President Obama missed a real golden opportunity, because he has a stable family, loving children -- great, balanced children from what I can tell -- a strong relationship with his wife, and he could have shown that as a model for others to emulate. And I don't think he's done it as much," Bush said.
Bush, as he often does, said that marriage is one of the keys to keeping people out of poverty, pointing to federal welfare programs that he argues create incentives for single moms to stay single, because otherwise they may lose some benefits.
The former Florida governor laid out parts of his proposal to reform welfare, but also argued that the "public persona of the presidency has power" on this issue as well.
"Maybe he's a private person, but I think that's important," Bush said. "I think how you lead your life as a president is as important as a five-point plan you have to deal with whatever subject it is."
Bush said that a family with a husband and wife provides the best environment to allow children to succeed.
"There may be other forms that are equally as important and good. I'm not passing judgment," he said. "To not try to strengthen family life, we do it at our peril."
Bush did not elaborate on what Obama could be doing to better illuminate his own family life. While his two daughters occasionally appear at public events, Obama and the first lady regularly talk about their family life in interviews.
"They're wonderful girls," Obama said in an interview this week with Ellen DeGeneres. "Michelle gets all the credit. Maybe an assist from my mother-in-law. But they really are just solid kids. They don't have an attitude. They're courteous and kind to everybody. They work hard. They don't feel like they're entitled to anything."
Bush has recently highlighted his family more on the trail, speaking about his daughter's drug addiction and arrest. In New Hampshire, where drug addiction has become an epidemic, Bush started sharing his daughter's story as a way to connect with audiences.
"Over time, Columba and I have realized that this is an important story to tell," Bush said in January on "Meet the Press" when asked why he started talking about it now. "And a lot of other people are going through this that aren't political. And it's just -- I think it's important to tell the story. This is one of the soft underbellies of our country. Addiction is a huge problem. And talking about it in a way that gives people a sense that it shouldn't be a stigma I think is important."