No, I don't think so. China shouldn't be the one start the first shot. After all, the Philippine side only tried to get away (even if they bait the Chinese side) and it is the incompetence on the Chinese side that cause them to crash their own ship.
fire first shot and what ? if you talk about firing first shot again someone actually can fight back or valueable enough for US to "must fire back or lose credit/face" like japan korea taiwan india australia then that can be understand
but this is philipines , the only thing they depend on is this
if you actually read it even NATO arctice 5 you will see the word used in this kind of paper treaty very : flexible
instead of clear cut words guarranted said other side must use military force to engage combat helping other party under attack
remind you : with treaty like this using correct words is very serious matter because if the words written in flexible way then the value of treatly gonna drop down by alot because it basically allow both party twist the those words meaning toward their benefit when the time require helping other side come
in case the confict they force to join in because of treaty not easy winable .
meaning : as long as there are no clear cut words force US fighting china or any other not-easy target to help allied , US can twist the meaning of "help" however their want sending some outdate military scrap aid , donate some money and bye also count as help ...
there are actually no clear cut words guarranted force US hand to use military force at all unless they actually want to
anyway here is the clip of a super pro-US propagrandist think tank guy actually admit even NATO arctice 5 was written in a very flexible way ...it never mean to be some kind of magic seal that when active instantly summon mighty US military in at all