In a 1951 playoff, Branca surrendered a walk-off home run to Bobby Thomson of the New York Giants; the game-winning hit was known as the "Shot Heard 'Round the World".

His mother, Kati (née Berger), who was Jewish, immigrated to the United States in 1901 from Sandorf, Hungary (now Prievaly, Slovakia).

[2][7] He debuted in the major leagues at age 18 on June 12, 1944, and put up a 3.04 earned run average (ERA) in 1092⁄3 innings pitched in his rookie year.

[7] In the final game of the best-of-three 1951 National League tie-breaker series at the Polo Grounds against the crosstown rival New York Giants, Branca entered the game in relief of Don Newcombe in the bottom of the ninth inning with one out and Whitey Lockman on second base and pinch runner Clint Hartung on third base and surrendered a walk-off home run, which became known as the "Shot Heard 'Round the World" to Bobby Thomson, giving the Giants the pennant.

Dodgers coach Clyde Sukeforth noticed that Erskine was bouncing several curveballs in the dirt and instructed manager Charlie Dressen to call on Branca—this despite Thomson having homered off Branca in Game 1.

Al Corwin, another member of the pennant winning Giants, also reminded others that "not many guys relied on it, because that pitcher isn't standing still, he's in the middle of his windup."

[2] In the eighth inning of Game 7 of the 1952 World Series against the New York Yankees, Branca was ejected from the dugout by home plate umpire Larry Goetz for bench jockeying.

[14] Branca became friends with Thomson, a bond that lasted into each man's old age, including joint television and trade show appearances.

[5][9] Branca later learned from Detroit Tiger Ted Gray that the Giants had stolen the signs to the two pitches he threw Thomson.

That rumor was confirmed in The Wall Street Journal in 2001, when Giant Sal Yvars admitted that he relayed to Thomson the stolen signs for Branca's fastballs.

[15] Joshua Prager detailed the revelations in a book entitled The Echoing Green: The Untold Story of Bobby Thomson, Ralph Branca and The Shot Heard Round the World.

Thomson acknowledged to Prager that the Giants had stolen signs in 1951, but denied that he had foreknowledge of the pitch he hit off Branca for the pennant-winning home run.

[20] In the 2012 movie, Parental Guidance, Branca made a cameo appearance as a judge during a scene involving an audition for a music school.

During the scene, Billy Crystal's grandson takes to the stage and recites the radio broadcast of, "The Shot Heard Round the World.

[22] Branca married Ann Mulvey, whose parents were part-owners of the Dodgers, in 1951, shortly after giving up the famous home run.