What is Weld to Weld minimum distance required according to ASME Code?

NO, there is not a minimum distance. A good practice is to bridge the weld of the vessel where it crosses with the stiffener ring.

Also, please, check UG 82 and UW 13 in ASME VIII. There are others ways to place this attachment.

you may want to look though the code book. the information you need should be in there