In the first version of the six-point text, seen by AFP, council members expressed "deep concern at the recent incidents" during a flare-up between the arch-foes across the "Blue Line" border.
The draft, drawn up by France, added that "members of the security council condemned all violations of the Blue Line, both by air and ground, and strongly calls upon all parties to respect the cessation of hostilities".
According to diplomats, Washington blocked the statement twice, calling for Hezbollah to be specifically condemned in the text.
Washington said it was impossible for it to back any statement putting Israel's right to self-determination on an equal footing with Hezbollah, which it considers a "terrorist organization", a diplomat explained.
Several other members of the security council objected to the US stance, and the text was eventually abandoned.
Any statement by the council must be backed by all 15 members.
Hezbollah, the Iranian-backed Lebanese movement, alleged Israel launched a drone strike on its Beirut stronghold late last month.
Lebanese President Michel Aoun denounced it as a "declaration of war".
Israel has not acknowledged that attack, but accused Hezbollah and Tehran of colluding to produce precision-guided missiles on Lebanese soil.