Ranatunga, 40, who was elected to Parliament in the April 2 election, said he entered politics because President Chandrika Kumaratunga invited him to be in charge of sports, but instead he was offered a junior position.
"I did not enter politics to make a name for myself," said Ranatunga. "The President clearly invited me to join her party and make a contribution by taking over the sports ministry.
"By appointing someone else, it shows I have no role to play (in politics). I have resigned as the joint national organiser of the youth wing of the SLFP (Sri Lanka Freedom Party)."
Kumaratunga appointed a close relative, former film star Jeevan Kumaratunga, as the sports minister in the 31-member cabinet.
The SLFP is the main constituent party in Kumaratunga's Freedom Alliance which narrowly won the election and formed a minority Government on Saturday.
Ranatunga, a hero in Sri Lanka after leading its cricketers to victory in the 1996 World Cup, said he was offered but declined the position of deputy minister of industries, tourism and investment promotion.
"It was a good position offered to me, but that was not what I was supposed to be doing. I am not keen on just getting some position. I don't want positions. All I wanted to do was contribute to improving sports."
However, Ranatunga said he will remain a backbench MP from the Freedom Alliance.
He was a key figure in Kumaratunga's SLFP forging an alliance with the Marxist JVP, or People's Liberation Front, to contest elections and oust the United National Party of former premier Ranil Wickremesinghe.
Ranatunga's father, Reggie Ranatunga, a former minister, lost the April 2 election after contesting in Kumaratunga's home constituency of Gampaha district.
Kumaratunga's brother Anura Bandaranaike was pushed to second place in Gampaha by a JVP member, in an embarrassing blow to the Bandaranaike dynasty which had produced three former Prime Ministers, a President and a Parliamentary Speaker.