Does doubt concerning a story related to the Prophet David (peace be upon him), and the saying of the Messenger (peace and blessings of Allah be upon him): "Indeed, David's sin was gazing," and the subsequent whispers about the infallibility of the Messenger (peace and blessings of Allah be upon him), expel a Muslim from Islam?

The feeling of doubt regarding the purity of the Prophet's heart, peace and blessings be upon him, is a demonic whisper (waswas). The mentioned hadith about the Prophet, peace and blessings be upon him, sitting with an beardless youth from the delegation of Abd al-Qays is a rejected (munkar) hadith. More than one scholar has ruled it to be rejected and fabricated, including Sheikh al-Islam Ibn Taymiyyah, al-Shawkani, al-Albani, and al-Zarkashi. The best cure for whispers is to disregard them and pay them no mind.