Is a man sinful if he divorces his second wife to appease his first wife, who demanded the divorce because she did not accept his marriage to another, and is the first wife entitled to demand a divorce in this situation given her feeling of injustice?

You bear no sin for divorcing your second wife for the reason mentioned. Divorce, when there is a need for it, is permissible. Your marriage to the second wife in the manner described does not constitute injustice to your first wife, nor is it a harm that permits her to seek divorce. However, if she cannot tolerate remaining with you after your marriage to the second wife, then she has the right to seek khul' from you.