 Symonds has been scoring plenty of runs in India |
Cricket Australia claimed Symonds was subjected to monkey noises during their one-day win in Vadodara on Friday.
Although Australia decided against taking further action, Ponting wants the International Cricket Council to enforce its strict anti-racism code.
"I know the match referee knows about it, if the ICC gets it in the report they have to do something," he said.
The Australians alleged a section of the crowd had made noises at Symonds while he was fielding on the boundary during the nine-wicket victory over India.
"Racism is unacceptable anywhere in the world. You don't expect it to happen when you step out on the field," said Ponting.
"We did not deserve it and we don't expect it to happen again."
The alleged abuse has been mentioned in match referee Chris Broad's report and it is expected the ICC will request more information on whether the Board of Control for Cricket in India (BCCI) has adopted the anti-racism code, implemented in November last year.
ICC members can impose a range of punishments on spectators, including a life ban, if they are found guilty of racial abuse, or withdraw international status from the venue where any incident took place.
BCCI secretary Niranjan Shah said local authorities in the state of Gujarat rejected claims that racial taunts were directed at Symonds.
"The police of Vadodara who were looking after the situation have denied this story," Shah told AFP.
"The code was implemented at the ground as per guidelines."
Australia, who lead the series 3-1, face India in the fifth international in Nagpur on Sunday.
Following a week that also saw accounts of racist abuse towards South Africa players and officials in their Test match against Pakistan in Lahore, ICC chief executive Malcolm Speed said both the Indian and Pakistan boards will be consulted over the reports.
Speed said: "The ICC retains a zero tolerance to racism, as illustrated in our anti-racism code - which was strengthened last year following consultation with our members and the ICC board.
"The code outlines measures host members are expected to undertake, and those measures were unanimously approved by the ICC's board.
"We need to understand whether those measures are working. If they are not then our members need to tell us why so we can ensure cricket remains free from the scourge of racism."
Tum Se Tum Tak
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