The 34-year-old is contesting the Gujarat polls from Vadagam in Banaskantha district as an independent candidate with outside support of the Congress.
The constituency, around 200km from here, is reserved for SC candidates.
Police said one of the vehicles in his convoy was hit by a stone, damaging its window, but no one was injured.
"A vehicle in Mevani's convoy was hit by a stone but nobody was injured. Mevani was in another vehicle. We will initiate whatever legal proceedings are required in the matter," said Banaskantha SP Niraj Badgujar.
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In another tweet, he asked Prime Minister Narendra Modi if it was "your idea or (BJP chief) Amit Shah's to attack those who are winning election, because this is not Gujarat's tradition".
However, the BJP denied Mevani's allegations.
Gujarat BJP spokesperson Jagdish Bhavsar said his party had nothing to do with the attack on Mevani's convoy.
Mevani is pitted against Vijay Chakravarthi of the BJP. The seat is currently held by Manibhai Vaghela of the Congress.
The Congress had asked Vaghela not to contest from the seat "as part of an agreement" with the Dalit leader.
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