In her message of condolence to current US President Donald Trump and the American people, the Queen described Mr Bush Snr as "a great friend and ally of the United Kingdom" who was a "patriot, serving his country with honor and distinction in Office and during the Second World War".
With hands on hearts, members of Bush's cabinet, lawmakers and Supreme Court justices honoured the 41st president as his flag-draped casket entered the Capitol Rotunda. President from 1989 to 1993, Bush died Friday at 94.
The Capitol will be open for dignitaries and the public to pay their respects through early Wednesday, with visitors allowed to walk past Bush's casket.
Bush spokesman Jim McGrath tweeted a photo of the presidential plane on a tarmac Sunday afternoon.
"After Mrs. Bush's death, general companionship was a big part of Sully's job", John Miller, president and CEO of America's VetDogs, said in a phone interview.
A funeral service has been planned for Wednesday at the Washington National Cathedral. "Bush of course was defeated for re-election and most people at the time considered him a failed president" because of a recession during his time in the White House.
Family members, former Bush cabinet members, and senators and representatives gathered to mourn Bush.
Memorial services began Monday to honor the late President George H.W. Bush.
"We'll be spending three days of mourning and three days of celebrating a really great man's life", Trump said in Argentina where he was attending the G20 summit.
Vice-President Mike Pence said there was a "kindness about the man that was evident to everyone who met him".
The casket will then be flown back to Houston, where the former head of state will lie in repose at St. Martin's Episcopal Church - where the Bushes worshiped for decades - until a funeral service at 1700 GMT Thursday.
Bush's service dog, a young Labrador retriever named "Sully", also walked past the casket and sat near it briefly.
Trump has ordered the federal government to close on Wednesday and both the New York Stock Exchange and Nasdaq will be closed in observance. He would go on to serve as ambassador to the United Nations and China, head of the Central Intelligence Agency and chairman of the Republican National Committee before being elected to two terms as Ronald Reagan's vice president. Flags on public buildings are flying at half-mast for 30 days out of respect for Bush.