No Duty 

Article II, Section 2 does not lay out any specific procedure by which the Senate can refuse its consent. It does not indicate whether it must do so by taking a vote, or whether it can simply refuse to consider the president’s nominee at all. However, Article I, Section 5 states that “Each House may determine the rules of its proceedings.” That power includes the rules for considering judicial nominations, as well as all other Senate business. Thus, so long as the Senate has established rules that allow it

Via The Constitution does not require the Senate to give judicial nominees an up or down vote @  Washington Post Volokh Conspiracy.

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