If worded so a majority winner based on honest voter full-rankings of all candidates must be elected when one exists and whenever the votes are consistent with those secret rankings, then Approval Voting fails MC.
If worded so a majority winner based on "the preferences deducible from the votes themselves" must be elected when one exists, then Approval obeys MC.
At one point the Wikipedia definition of MC was "if a majority of voters ranks a given candidate higher than every other candidate, then that candidate should win." Note that Approval obeys MC under this wording. Wikipedia then, in order to prevent Approval Voting from satisfying MC introduced a special addendum to their wording: "For non-ranked ballots (e.g. Approval voting or Range voting, it can be expressed as follows: 'If more than half of the voters give candidate X a higher rating than any other candidate, the winner should be candidate X'."
Range Voting does satisfy a different, and in our opinion more important, form of the Majority Criterion: "If a majority of voters regard X to be the best candidate, then that majority has a simple and obvious way to force X's election."
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