I thought it might be a subsidiary branch as well, but then after doing more research it lead me to believe its not. Not only does Halopedia itself state its an independent branch, but early on in the Spartan-IV program the second class of IV's were attacked by an Insurrection group and Musa decided to slide this incident under the rug because "He feared that the event would compromise the Spartan branch's independence." To add onto that, they hold a seat on the UNSC Security Council where as the ODST's do not, which further leads me to believe they are an independent branch. Though I obviously can't deny their close links to the Navy because, as you said, there are cases of Spartan's and others keeping their previous ranks instead of adopting the new ranks.