Back in those days it was the responsibility of the each league to make these decisions. I will cite the case below as a reference for you(I added the emphasis):
"Brett had injuries on-and-off for the next four years, during which occurred the most notable event in his career, the notorious "Pine Tar Incident". On July 24, 1983, the Royals were playing the Yankees at Yankee Stadium. In the top of the ninth inning, Brett came up to bat against Goose Gossage, his old rival. Brett hit a two-run homer to put the Royals up 5-4. After Brett rounded the bases, Yankees manager Billy Martin calmly walked out of the dugout and used home plate to measure the amount of pine tar, a legal substance used by hitters to improve their grip, on Brett's bat. Martin cited an obscure rule that stated the pine tar on a bat could extend no further than 18 inches. Brett's pine tar extended about 24 inches. Earlier in the season, the Yankees had noted Brett's habit of adding pine tar further than the allowed 18 inches, but waited until a crucial time to point it out to the umpires.
"I've never seen this," said sportscaster and ex-Yankee Bobby Murcer on WPIX as he watched McClelland measure the bat across the plate. "I never have either," said Murcer's partner, Frank Messer. A few moments later, the home plate umpire, Tim McClelland, signaled Brett out.
The normally mild-mannered Brett charged out of the dugout, enraged, and was immediately ejected. An incredulous Messer:
“ Look at this!...He is out, and having to be forcibly restrained from hitting plate umpire Tim McClelland. And the Yankees have won the ball game 4 to 3! ”
Years later, Brett explained his outburst by saying "It was just such an extraordinary thing to hit a homer off [Gossage], the thought of losing it was too much". In the same interview he also humorously chided his teammate Hal McRae (who was on deck) for not removing the bat from home plate before Billy Martin could have it inspected. "If Hal had [taken the bat], then I'd only be known for hemorrhoids," Brett quipped.[3]
The Royals protested the game,
and their protest was upheld by AL president (and former Yankees chief executive) Lee MacPhail, who ruled that the bat was not "altered to improve the distance factor," and that the rules only provided for removal of the bat from the game, and not calling the batter out.
The game was continued later that season, starting after Brett's homer. Billy Martin had one last trick up his sleeve, appealing the play in saying the umpires had no way of knowing Brett and the other runner had touched all the bases. Martin was stunned when the umpires produced affidavits saying they had. The game had virtually no effect on 1983's pennant race, but was in many ways the closing chapter on a heated rivalry. The Pine Tar Game has become part of baseball folklore, with Brett's famous bat on display at the Baseball Hall of Fame in Cooperstown, New York."
If you are interested, you can read the article in its entirety
here.