。
She swore off drinking.
Literal
She [topic-は] alcohol [object-を] quit [quotative-と] swore.
誓う (ちかう) means to swear or vow — a strong commitment word. 酒をやめると誓った uses the quotative と to mark the content of the vow: 'she swore that she would quit alcohol.' やめる (to quit/stop) is the casual, direct verb for giving something up. The combination of the casual やめる with the solemn 誓う creates a natural contrast between the simple action and the weight of the promise.