Is it Islamically permissible for a partner to sell his share in a car, whose value was estimated at double the original amount he paid, in exchange for monthly installments, given that the other partner completed the remaining amount to purchase the car?

The ruling is based on the reality of the transaction. Lending a person the remainder of the price for him to repay it multiplied is impermissible because it is a usurious loan.

However, if you pay the remainder of the price to buy the car between yourselves, such that you own a share of it commensurate with what you paid, and then you sell him your share for an installment price, there is no harm in that, even if the price is higher than the market value. For selling with immediate payment is not like selling with deferred payment.

Therefore, the transaction must be conducted in this manner: that you become partners in buying the car according to what you both paid, then you sell him your share, or lease it to him, or enter into a diminishing partnership agreement with him.

The important thing is that the contract should not be merely a loan with interest, for selling with profit is permissible, and lending with interest is usury. Allah, the Almighty, says: "Those who consume interest will not stand [on the Day of Resurrection] except as one stands who is being beaten by Satan into insanity. That is because they say, 'Trade is [just] like interest.' But Allah has permitted trade and has forbidden interest." (Al-Baqarah: 275).