The purpose of the Kepler spacecraft launched by NASA is option 4, which is to search for Earth-size planets moving around other stars.

Option 1, which is to determine the existence of water or signs of conditions favorable to the formation of water on the moon, is not correct as this is not the main objective of the Kepler spacecraft.

Option 2, which is to repair the Hubble Telescope already positioned in space, is also not correct as the Kepler spacecraft is a separate mission from the Hubble Telescope and does not involve repairing it.

Option 3, which is to look for signs of organic molecules or conditions leading to their formation on Mars, is also not correct as the primary focus of the Kepler spacecraft is not on Mars or the search for organic molecules.

Therefore, the correct answer is option 4, as it accurately describes the purpose of the Kepler spacecraft – searching for Earth-size planets orbiting other stars.