Is the Isha prayer acceptable if the worshipper doubts having said the takbir when moving from the first tashahhud to the standing position, knowing that he completed his prayer?

Your prayer is valid and what you did was correct. Those who mandate prostration of forgetfulness for omitting a takbir (other than the takbirat al-ihram)—namely the Hanbalis—do not require prostration unless forgetfulness is certain, not merely suspected. As for the Shafi'is, they do not see any prostration of forgetfulness for omitting a takbir at all. The Malikis, on the other hand, only require prostration for omitting three takbirs or more due to forgetfulness.