Does a divorce issued by a husband to his wife with the words "you are divorced" occur after a quarrel that resulted in him pushing her, her head hitting the wall, and her spitting on him?

Raising children is a shared responsibility of both parents, especially the father. The proof for this is Allah's (the Exalted) saying: "O you who have believed, protect yourselves and your families from a Fire." (Quran 66:6). They should be dealt with wisely and gently, not harshly, as confirmed by the Prophet's (peace be upon him) saying: "Gentleness is not found in anything except that it adorns it, and it is not removed from anything except that it disgraces it."

Regarding your statement to your wife: "You are divorced," a divorce occurs with it, whether you intended it or not, because an explicit divorce does not require an intention. Anger is not taken into consideration unless it is so severe that you do not know what you are saying. If the divorce occurred and it was a revocable divorce (first or second) and she is still within her waiting period (iddah), you can take her back without a new contract or mahr (dowry). We advise you to take her back for the sake of the children.