The president said he would name a successor to Associate Justice Sandra Day O'Connor, who announced her retirement at the end of the high court's last term, in a "timely manner."
Roberts was originally named as O'Connor's replacement and awaited Senate confirmation hearings starting Tuesday.
Senate Judiciary Committee Chairman Arlen Specter, R-Pa., is reportedly considering postponing those hearings amid calls for a delay by key Democrats.
Rehnquist succumbed to thyroid cancer at his Arlington, Va. home Saturday morning at age 80. He served more than 33 years on the high court and had been chief justice since 1987.
"The passing of Chief Justice William Rehnquist leaves the center chair empty, just four weeks left before the Supreme Court reconvenes," the president said. "It's in the interest of the court and the country to have a chief justice on the bench on the first full day of the fall term."
Filling the position of chief justice quickly avoids having liberal Justice John Paul Stevens deciding to whom to assign cases and other decisions that could influence court deliberations, according to the Associated Press.
President Bush urged the Senate to act within the next few weeks to confirm Roberts to the Supreme Court's top position before the court's next term begins Oct. 3.
"He's a man of integrity and fairness and throughout his life he's inspired the respect and loyalty of others," the president said of Roberts at the announcement in the Oval Office. "John Roberts built a record of excellence and achievement and reputation for goodwill and decency toward others in his extraordinary career."
"I am honored and humbled by the confidence the president has shown in me," Roberts replied. "I am very much aware that if I am confirmed I would succeed a man I deeply respect and admire, a man who has been very kind to me for 25 years."
In 2001, President Bush nominated Roberts to the U.S. Court of Appeals for the D.C. Circuit. Roberts was unanimously confirmed by the Senate in 2003.
Roberts at one time clerked for Rehnquist, and has argued 39 cases before the high court in the public and private sector.
Liberal groups have come out against Roberts' nomination due to his conservative writings as an attorney for the Reagan administration. However, it does not appear opponents have enough votes to block his confirmation, reported the AP.