TL;DR: Why not White Phosphorus?
— Because it appears to be easier to convince the world community by condemning Assad regime's use of other chemical weapons — namely, chlorine, sarin, and mustard gas.
This gives the U.S. and allies (¹) some diplomatic freedom, effectively helping their argument to justify the need of overthrow the Assad's regime, which, in turn, is the final goal for the U.S. in the region.
(¹) if I correctly understand the term of "World" in the question
As other users noted, the U.S. established a "red line" at the use of chemical weapons (CW) in general. No specific chemical substances were named.
Indeed, if the U.S. argument were Assad regime's use of White Phosphorus (WP), Assad/Russia lobby would receive a powerful counter-argument, raising a long-running debate:
- about whether the U.S. and allies have also used it in their operations;
- or whether or not WP should be considered a banned CW;
- or whether WP was only used to illuminate the battlefield (which is arguably a legal use).
Yesterday's (24-Aug-2016) statement by NSC Spokesperson Ned Price on the UN-OPCW Report on Syria sums it up very well (markup is mine):
It is now impossible to deny that the Syrian regime has repeatedly used industrial chlorine as a weapon against its own people in violation of the Chemical Weapons Convention and UN Security Council Resolution 2118. We condemn in the strongest possible terms the Asad regime's use of chlorine against its own people.
The United States will work with our international partners to seek accountability through appropriate diplomatic mechanisms, including through the United Nations Security Council and the OPCW.
- The final goal for U.S. in Syria is overthrowing the Bashar al-Assad regime.
- However, "I don't think we should remove another dictator with force." — President Obama
- The remaining way to accomplish the goal is by diplomatic path — Ned Price;
- Which requires solid, convincing arguments;
- The use of chlorine appears to be such an argument.