Is one who tried to rescue a drowning person, and thought he caused his death by an unintentional movement during the rescue attempt, sinful, and is his action considered accidental killing that necessitates expiation?

It is obligatory for anyone capable to rescue a drowning person, even if they are performing prayer or fasting. One who fails to rescue them is sinful. As for the ruling on one who neglects to rescue them, the Hanafis, Shafi'is, and Hanbalis held that they are not liable for compensation, though they are sinful. The Malikis and Abu al-Khattab from the Hanbalis held that they are liable.

If a rescuer begins to rescue someone, and the drowning person clings to them, and the rescuer fears for their own life, they may abandon them, in accordance with the verse: "And do not throw [yourselves] with your [own] hands into destruction."

If the rescuer pushed the drowning person unintentionally, leading to their drowning, there is no sin upon the rescuer. However, experts should be consulted to determine whether this push was the cause of the drowning. If it was, then expiation is incumbent upon the rescuer, and the blood money (diyah) is upon their 'aqila (male agnatic relatives).