Does the prayer become invalid if the worshiper, out of ignorance, makes the tashahhud gesture with his left hand, even though the correct way is to do so with the right hand?

If you are referring to pointing with the index finger during the tashahhud, the Sunnah is to point with the right index finger, while leaving the left hand flat with the fingers close together. Ibn Qudamah stated in "Al-Mughni" that the worshiper places his left hand on his left thigh, spread out with fingers together, and places his right hand on his right thigh, clenching the pinky and ring finger, forming a circle with the thumb and middle finger, and pointing with the index finger. It is disliked to point with both index fingers because the Sunnah for the left hand is to remain spread out. Your omission of this Sunnah due to ignorance does not invalidate your prayer, but in the future, you must point with your right index finger during the tashahhud.