After the veto, the council unanimously adopted a measure that extended for one year the sanctions regime against Yemen, but that text made no mention of Iran.
The Russia veto was a setback for the USA as it pushes the Security Council to take a stronger stance towards Iran over its role in regional conflicts and missile tests.
The West and its regional allies have accused Iran of violating the United Nations arms ban on the militants in Yemen.
Both resolutions were meant to renew the sanctions imposed on Yemen, which were set to expire on Monday.
The UN's independent panel of experts, which monitors the implementation of sanctions, concluded in its most recent report that Iran had failed to prevent weapons transfers to the Houthis.
His remarks came a day after Russian Federation vetoed a UK-drafted resolution at the UN Security Council, urging for "additional measures" against Tehran over the alleged breach of the 2015 arms embargo on Yemen, having supplied weapons and military equipment to the Houthi revolutionaries.
The experts examined missile remnants fired into Saudi Arabia by the Houthis a year ago and said many "are consistent with those of the Iranian designed and manufactured Qiam-1 missile".
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France, Germany, the United Kingdom and United States issued a joint statement saying Iran's non-compliance, as described by a UN panel of experts, "poses serious risks to peace and stability in the region". And he warned that the resolution's adoption would have "dangerous, destabilizing ramifications" in Yemen and "inevitably escalate regional tensions and lead to conflicts among key regional players".
U.S. Ambassador to the UN Nikki Haley.
"Today, Russia protected the terrorist-sponsoring regime in Iran", said Haley.
An earlier version of the draft included a condemnation of Iran for failing to block the military supplies to the Huthis, but that was dropped from a final version that Russian Federation nevertheless vetoed.
Iran's mission to the United Nations accused the USA and Britain of pushing an "unwarranted draft resolution" to advance an anti-Iran political agenda and "categorically" rejected allegations of arms transfers to Yemen.
"Iran has spiritual and emotional ties with Houthi Ansarullah movement and politically support them to help Yemen secure its sovereignty", he added, asked about Tehran's relations with the Houthis. You can sign up to receive it directly here.