First, it seems this was not a real/open veto, but what is called a hidden veto, i.e. one or more Permanent members of the UNSC expressing disagreement on the wording, so that no actual voting takes place.
Unlike some Turkish sources, the Washington Post does not mention a veto having been exercised, but does note that both the US and Russia disagreed with the wording proposed by the Europeans, although apparently for different reasons:
Five European ambassadors who had called the meeting hoping to present a unified front against Turkey stood together with a sixth, from Estonia, and demanded that Turkey cease its military operations. [...]
Kelly Craft, the U.S. ambassador to the United Nations, said the Trump administration does not endorse Turkey’s military action and warned of unspecified “consequences” but stopped short of condemning it.
“Failure to play by the rules, to protect vulnerable populations, failure to guarantee that ISIS cannot exploit these actions to reconstitute, will have consequences,” she said, using a common acronym for the Islamic State.
The Russian U.N. ambassador, Vassily Nebenzia, accused the United States and its coalition allies of conducting “demographic engineering” that he said led to the conflict. He called for a solution that would “take into account other aspects of the Syrian crisis, not just the Turkish operation.”
“It should speak about the illegal military presence in that country,” he said in an apparent reference to U.S. troops in Syria.
Slighly more detail from a Canadian source on what the US would have preferred.
The United States proposed a statement that expressed "deep concern," called for protection of civilians, and asked Turkey to go through diplomatic channels rather than take military action, council diplomats said, speaking on condition of anonymity because consultations were private. [...]
So the Europeans issued their own statement after the meeting, urging Turkey "to cease the unilateral military action." They said the offensive threatens progress against the extremist Islamic State in Iraq and Syria (ISIS) by a global coalition, undermines stability of the region and exacerbates "civilian suffering."