No, the song "It Might Be You" from the movie Tootsie did not win an Oscar. However, it was nominated for the Academy Award for Best Original Song in 1983. The song was composed by Dave Grusin with lyrics by Alan and Marilyn Bergman.
The Jazz Singer
Love Is a Many-Splendored Thing won the Oscar for Music - Song - in 1955.
Breakfast at Tiffany's won the Oscar for Music - Scoring - in 1961.
Fame won the Oscar for Music - Song - in 1980.
Gigi won the Oscar for Music - Song - in 1958.
The Jazz Singer
The Best Original Song award went to "When You Believe" from "The Prince of Egypt."
Song of the South won the Oscar for Music - Song - in 1947.
Love Is a Many-Splendored Thing won the Oscar for Music - Song - in 1955.
"State Fair" (1945). He shared the Best Original Song award for "It Might As Well Be Spring" with his longtime collaborator Oscar Hammerstein II.
Breakfast at Tiffany's won the Oscar for Music - Scoring - in 1961.
Aladdin won the Oscar for Music - Song - in 1992.
Evita won the Oscar for Music - Song - in 1996.
Fame won the Oscar for Music - Song - in 1980.
Flashdance won the Oscar for Music - Song - in 1983.
Gigi won the Oscar for Music - Song - in 1958.
Once won the Oscar for Music - Song - in 2007.