WASHINGTON—President Barack Obama has supported the democratically elected governments that have emerged in the Mideast over the past 18 months, despite concerns about the influence of Islamists that historically opposed the U.S.

Tuesday's attacks put that policy at a crossroads. On Wednesday, some American lawmakers and policy makers called for the U.S. to suspend financial aid, and possibly security cooperation, to Egypt and Libya, countries that were transformed by the revolutions of the Arab Spring.

As...