The couple landed in the grounds just before 17:00, a few minutes ahead of schedule, and were greeted by the Queen in the castle's Quadrangle.
A Guard of Honour formed of The Queen's Company First Battalion Grenadier Guards gave a royal salute and the US national anthem was played.
Mr Biden accompanied the Officer Commanding the Guard of Honour, Major James Taylor, and Major General Christopher Ghika, to inspect the Honour Guard, before returning to the dais to watch the military march-past with the Queen and First Lady.
The Bidens walked into the castle with the monarch via the Sovereign's Entrance.
Mr Biden walked alongside the Queen, who said to the President: "You completed your talks."
He replied: "Yes we did."
Before going inside, the Queen introduced the Bidens to her lady-in-waiting Dame Annabel Whitehead.
Mr Biden, who is in the UK for the G7 summit, will be the 13th American leader to meet the monarch, with Lyndon B Johnson the only one the Queen has not met.
The tea in the castle's Oak Room comes after the Bidens met the Queen on Friday when she attended a reception for G7 leaders at the Eden Project.
Escorting a visiting head of state to inspect the troops is a role that usually fell to the late Duke of Edinburgh before he retired in 2017.
The carefully choreographed arrangements to welcome Mr Biden echo Donald Trump's visit in 2018 when the controversial then-US president travelled to Windsor to meet the monarch.
Mr Trump and the Queen had afternoon tea together, also in the Oak Room, with the meeting lasting longer than expected.
It was due to last about half an hour but overran by almost 20 minutes.
On that occasion, however, the Queen joined Mr Trump in inspecting the troops.