The horse, named Shaktimaan, suffered multiple fractures and some reports said it might require a leg amputation.
Mr Joshi said the charge against him was politically motivated.
"The footage which the electronic media is showing by linking with an old clip - in which I am lifting a stick in front of the horse - has no connection at all [to the incident]," Mr Joshi told the BBC.
"The horse fell down when someone pulled its saddle and the animal sustained a fracture on its leg," he added.
The incident took place near the legislative assembly in the state capital, Dehradun, on Monday when the BJP was holding a protest against the Congress party-led state government.
The horse is being treated by veterinary officers and reports say it is in a serious condition.
The incident has caused outrage in India with many taking to social media to express their anger.
On Twitter, the hashtag #IsHorseAntiNational, prompted by a tongue in cheek newspaper headline that read, "BJP MLA beats up a horse. No confirmation whether the horse was anti-national", has been trending since Tuesday morning.
The term "anti-national" is being used in reference to the recent arrest of several students of a top university for allegedly shouting "anti-India" slogans.
The government defended the arrests by saying the students were "anti-national".
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