The 81-year-old Democrat posted on X that he would remain in his role as president and commander-in-chief until his term ends in January 2025.
He endorsed Vice President Kamala Harris to replace him as his party's candidate in the November contest against former president Donald Trump.
Early on Monday, Mr Albanese posted a message on X expressing his gratitude to Mr Biden.
"Thank you for your leadership and ongoing service President Biden," Mr Albanese wrote.
"The Australia-US Alliance has never been stronger with our shared commitment to democratic values, international security, economic prosperity and climate action for this and future generations."
Thank you for your leadership and ongoing service President Biden. The Australia-US Alliance has never been stronger with our shared commitment to democratic values, international security, economic prosperity and climate action for this and future generations. — Anthony Albanese (@AlboMP) https://t.co/gkAggLguAjJuly 21, 2024
Australia's ambassador to the US Kevin Rudd, whose prime ministership fell victim to a leadership spill in 2010, noted the difficulty of Mr Biden's choice.
"Leadership can be a lonely place," he posted on X.
"We can all take pause to recognise and respect the difficult decision President Joe BIden has made today.
— Kevin Rudd AC (@AmboRudd) pic.twitter.com/pC1j2tZvJgJuly 21, 2024
"Thank you for your service, Mr President, and for always being guided by what you believe is best for your country."
Opposition Leader Peter Dutton thanked Mr Biden and highlighted the importance of a security alliance between the US, UK and Australia that the American president helped launch.
"Like American Presidents before him, Joe Biden has been a leader resolutely committed to the alliance," he posted on X.
"I thank the President for the support and abiding friendship he has shown to Australia.
On behalf of the Coalition and Federal Opposition, I thank Joe Biden for his service and leadership as the 46th President of the United States. — Peter Dutton (@PeterDutton_MP) https://t.co/EY4IuXPH6WJuly 21, 2024
"In particular, I thank President Biden for launching the defence endeavour of AUKUS which the peoples of our two countries carry forward with energy and determination."
Mr Biden wrote that it had been the greatest honour of his life to serve as US president.
"And while it has been my intention to seek re-election, I believe it is in the best interest of my party and the country for me to stand down and to focus solely on fulfilling my duties as President for the remainder of my term."
Mr Biden's announcement follows public and private pressure from Democratic lawmakers and party officials to quit the race after his poor performance in a June 27 televised debate against the 78-year-old Republican rival.
The resignation has come as a "shock, not a surprise", University of Sydney United States Study Centre fellow Bruce Wolpe said.
Following the debate, Mr Biden's position was "completely untenable".
"It was a catastrophe, as everyone has seen, his position was politically terminal," he told ABC radio on Monday.
Democrats lost confidence in the president's ability, prompting Mr Biden to bleed support from members of Congress who felt that "the whole of Washington was at risk".
with Reuters