No you must pass the stabilized gate well before minimums; legal requirement is usually 1000ft above the airport for IMC (instrument conditions), 500ft for visual conditions. AFAIK airlines can set higher minimums and may set other configuration gates (eg 1500ft gear must be out, flaps in landing configuration).
To be stabilized you must be lined up laterally (eg localizer captured) and on glide slope (vertically aligned). Airspeed must be Vref +/-10, final landing configuration (flaps, gear, etc), sink rate correct (<1000fpm), power set (not idle), and all checklists complete.
Typically the pilot monitoring should make a callout for this (eg "Approach") and the pilot flying responds ("stabilized" or "unstable, going around").
The "minimums" callout is to verify that you can see the runway sufficiently to land. The exact requirements vary - a Category I ILS you can go down to 200ft without seeing the runway. A Category IIIb ILS is 50ft. Technically Category IIIc allows 0 so has no decision height and requires no "minimums" callout but I don't know of anyone crazy enough to allow this - you wouldn't be able to taxi off the runway with zero visibility.
The stabilized approach gate is a performance & configuration check. Is the aircraft configured for landing and on the correct performance profile with the expected automatic guidance?
Minimums callout (DA/MDA) is a visibility check. Can we see the runway to whatever degree our approach requires?
btw: If you are curious the various ILS categories impose performance and redundancy requirements on both aircraft and the airport ground equipment. Things like backup transmitters, self-monitoring to catch drift, battery backups, etc.