Does Islamic law approve of a man marrying another woman without his first wife's knowledge, despite his having signed a pledge not to marry anyone else while married to her?

If you stipulated to your husband that he should not marry another wife while you are his wife, and he has not violated this condition, then the fact that he was previously married and did not inform you is not a violation of Islamic law nor of the condition. Although it would have been better for him to inform you, it was not obligatory for him. What is obligatory upon him is to be just between you and to fulfill your rights concerning maintenance, clothing, and kind companionship. A Muslim woman must be content with Allah's حكم (judgment/rule), for what Allah has legislated is justice and mercy.