I don’t think so.  From the way you describe it, there was no vote by show of hands.  There was a voice vote and the secretary asked for a specific indication from those who voted no in order to determine whether there were enough negative votes to defeat those in favor.  When it became clear that there were not enough votes to defeat the motion, the motion was declared passed by the voice vote.

Robert’s Rules of Order, which have no force of law unless your bylaws have adopted them as governing principles, deal specifically with such a situation.  They provide that an inconclusive vote may be “verified” by a show of hands if no one objects.  If anyone objects, the vote can be redone by a standing division of the members.  The rules also make clear that if there is an objection to the validity of the vote, it must be made immediately and not at some subsequent meeting.

Those rules make sense, in trying to determine the will of the group at a time the group is still intact to make a final decision.  A board making a later determination merely frustrates the will of the group.

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Robert's Rules: QuickStart Guide - The Simplified Beginner's Guide to Robert's Rules of Order

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For well over a century-- from the hallowed halls of government to the executive boardrooms of Fortune 500 companies, to the meeting halls of labor unions--Roberts Rules of Order has been the how-to authority on applying organizational intelligence to deliberative assemblies. When properly utilized, Robert's Rules ensure that the best ideas, not just the loudest, are always allowed to surface. They ensure that problems identified are not forgotten, but resolved, and responsibilities are always clearly defined and never breached without consequence.