Would a husband be sinful if he married another woman after promising his wife that he would not marry another, and then regretted his promise?

Fulfilling a promise is highly recommended, and breaking it is strongly disliked. The Prophet, peace and blessings be upon him, considered breaking a promise to be one of the characteristics of hypocrisy.

If the promise was made by you, and you would be harmed by not taking a second wife, then there is no harm in marrying again. Strive to convince your wife. However, if your wife stipulated before the marriage contract that you would not take a second wife, and you accepted, then the most preponderant opinion is that this condition is valid and binding.