Bush will also lie in repose at St. Martin's Episcopal Church in Houston before a service there Thursday.
Celebrities, including actor-politician Arnold Schwarzenegger, were among the mourners at the church where the Bush family worships.
Prior to the start of the ceremony, the president used his preferred platform - Twitter - to send a message of remembrance, urging Americans to treat the occasion not as a funeral, but as a celebration of the former president's life.
Following the funeral at St. Martin's, Bush's casket will be taken by train to his presidential library at Texas A&M University in College Station.
Calling Bush a "charter member of the Greatest Generation", Baker said Bush was the "nation's best one-term president".
But at virtually every turn, in every rhetorical flourish hailing the elder Bush as a paragon of leadership, virtue and character, the contrast and the tension were palpable.
"Looking forward to being with the Bush family", he tweeted early Wednesday, before Bush's remains were transported from the Capitol to Washington National Cathedral. "They will really be watching President Bush, but Navasota will be seen across the world". "He stood in the breach in the Cold War against totalitarianism".
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In his first public meeting with Trump since the inauguration, Obama and Trump both cordially reached out to shake hands as Trump was seated.
Trump also shook Michelle Obama's hand and exchanged a greeting.
"I don't think it's respect for the human being if you talk smack about them, if you say the nastiest things about them, if you treat them as subhuman, and then you want to show up at their funeral".
Bush's service dog, Sully, was taken to the viewing, too - his main service these last months since Barbara Bush's death in April being to rest his head on her husband's lap. "They had just as much animus for Bush 41 and Bush 43 as they do Trump, for different reasons".