No way:

1) I justed dialed the number the OP called and did not get any such message before being connected to an agent (after going to the automated phone booking thing and saying "agent").
2) Generally those messages say something along the lines of "You can avoid this fee by booking online", which the OP attempted to do but was calling in to correct a problem with. At the very least, his situation creates enough ambigutiy where United couldn't presume "consent" for a fee since he'd already gone and done what they told him to do to avoid the fee.
3) If the reservations agent told him they were going to charge him $XXX (XXX being the price of the ticket) and did not inform him that they were going to charge him another $15, and he consented to be charged $XXX, no reasonable person would construe that to mean that he had actually consented to pay $XXX+$15. Generally, when I have paid phone booking fees the agent either incldes the $15 in the initial price they quote me or they tell me that there's also a $15 fee and tell me what the total is. That is the only acceptable way to gain consent for a charge when you have the person on the phone consenting to have the credit card fee. Tacking on an extra $15 "because we told you about it while you were waiting on hold" is the stuff that scams are made of, especially when the have the opportunity to tell you about the fee and gain your consent while booking your ticket.