Bush, 88, the nation's oldest living former president, was admitted to Methodist Hospital on Nov. 23. His stay included a week in intensive care last month.
"I am deeply grateful for the wonderful doctors and nurses at Methodist who took such good care of me," Bush said in a statement released by spokesman Jim McGrath. "Let me add just how touched we were by the many get-well messages we received from our friends and fellow Americans. Your prayers and good wishes helped more than you know, and as I head home my only concern is that I will not be able to thank each of you for your kind words."
Bush had been in the hospital for about a month before his office disclosed in late December that he was in intensive care because physicians were having difficulty controlling a fever that developed after the cough improved.
His office said on Dec. 29 that he had been moved back to a regular hospital room. Since then, his condition has continued to improve and he has been undergoing physical therapy to rebuild his strength.
"Mr. Bush has improved to the point that he will not need any special medication when he goes home, but he will continue physical therapy," Amy Mynderse, the doctor in charge of Bush's care, said in Monday's statement.
Bush's office said he was treated for a bacterial infection, along with the bronchitis and cough.
Bush and his wife, Barbara, live in Houston during the winter and spend their summers at a home in Kennebunkport, Maine. On Jan. 6, they celebrated their 68th wedding anniversary. They are the longest-married presidential couple.
Bush had served two terms as Ronald Reagan's vice president when he was elected in 1988 to be the nation's 41st president. Four years later, after a term highlighted by the success of the 1991 Gulf War in Kuwait, he lost to Democrat Bill Clinton amid voter concerns about the economy.
Bush has a long record of service, beginning with his enlistment in the Navy in World War II. At one point, he was the nation's youngest naval aviator. He was shot down in the Pacific and rescued by an American submarine. He's also been a congressman from Texas, U.S. ambassador to China and CIA director.
He suffers from a form of Parkinson's disease that has forced him in recent years to use a motorized scooter or wheelchair to get around.