So reports Abby Ohlheiser (Washington Post’s Checkpoint):

Following a review of the policy by the Department of Defense General Counsel, the Air Force will now permit airmen to omit the phrase, should they so choose. That change is effective immediately, according to an Air Force statement.

Exactly right, both as a matter of good sense and as a matter of law — the relevant statutory scheme expressly allows “affirm[ing],” which means making a “pledge equivalent to an oath but without reference to a supreme being or to ‘swearing.’”