The spell is countered. A cost of 0 is not paid automatically under the comprehensive rules (C.R. 117.5), and the "unless" formulation on Clash of Wills means that paying X is an optional cost (C.R. 117.12a, 117.12), so your opponent can choose whether the cost is paid.
Just note that, in a tournament setting, your opponent would have to be explicit about it (since 0 cost are paid automatically unless otherwise specified.)
Quote from MTR »
A player is assumed to have paid any cost of 0 unless he or she announces otherwise.
If the opponent says the spell is countered or move a permanent spell to the graveyard, it's the same as saying he doesn't pay the cost.
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He then decides not to pay the 0. Is my spell countered?
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EDIT (Feb. 15; Dec. 1, 2017; Apr. 5, 2018): Clarification.
Thanks
If the opponent says the spell is countered or move a permanent spell to the graveyard, it's the same as saying he doesn't pay the cost.