Actually MLB home team supplies balls for both teams, which has lead to it's own series of scandals and controversies as some teams accused others of doctoring balls or switching from livelier to deader balls between innings.
Poor wording on my part. Should have specified that unlike NFL, both MLB teams play with the same set of balls. In any given game, balls are (supposed to be) the same for all pitchers. The rules do not allow one set of balls for home team's pitchers and a different set of balls for visiting pitchers.
You are correct that before each MLB game the home team gives the umpire a "supply of regulation baseballs, the number and make to be certified to the home club by the league president." (Rule 3.01c). Umpire is tasked with inspecting them before the game.
http://mlb.mlb.com/mlb/downloads/y2014/official_baseball_rules.pdf
Of course MLB has had its' own ball (and bat) scandals. Always will be those trying to bend the rules