Probably. Unless the new president is the sole member of the corporation, which does not sound likely from the information you have given, he does not control the organization no matter who signed what and is subject to the basic rules of the nonprofit corporation law of your state. Bylaws should provide that the board has the power to remove officers (See Ready Reference Page: “Bylaws Function as ‘Constitution’ of Nonprofit Corporations”), but if they are silent on this issue, check the state statute. Most state laws generally provide that an officer may be removed by the board whenever in its judgment it would be in the best interests of the corporation. Since the statute controls where the bylaws are silent, if you have such a provision in your state law, you could vote him out of office.
It may be more difficult to fire him as a member of the board entirely, assuming that the bylaws are also silent on this point, since the statutes tend to place a much higher hurdle for terminating board membership. They usually require things such as conviction of a felony or a judicial declaration of mental incompetence. Once he is removed from the presidency, however, he may realize he isn’t wanted and resign from everything. If he doesn’t, you may be able to amend the bylaws, like those in the Ready Reference Page, to give the board such authority. At the worst, you tolerate him now but don’t reelect him as a director when his current term is completed.