B737 DEP FROM RDU CONTINUES TKOF AFTER TWR ISSUED ABORT; ACFT ON FINAL EXECUTES GAR.

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DURING TAXI OUT FOR DEP; GND CTL ASKED IF WE WOULD BE READY AT THE END FOR TKOF. I TOLD GND CTL WE'D NEED ABOUT 30 SECONDS. AS I SWITCHED TO TWR FREQ I WAS FINISHING UP THE BEFORE TKOF CHKS DOWN TO THE FINAL ITEMS AND CALLED TWR TO ADVISE 'READY IN SEQUENCE' TO WHICH WE RECEIVED A RETURN CALL SOMETHING LIKE; 'TFC 4 MI FINAL; FLY 035 DEG HDG; CLRED FOR IMMEDIATE TKOF.' AS I RECALL WE DIDN'T EVEN STOP SHORT OF THE RWY; BUT CONTINUED FOR A ROLLING TKOF AS THE BEFORE TKOF CHK WAS COMPLETED. WHEN ALIGNED WITH THE RWY; TKOF THRUST WAS APPLIED. AT ABOUT THIS POINT TWR ASKED US IF WE WERE ROLLING TO WHICH I RESPONDED 'YES; WE ARE;' WE HAD NEVER STOPPED. IT WAS SHORTLY AFTER THIS THAT TWR ASKED THE APCHING ACFT IF THEY WERE 'GOING TO MAKE IT;' TO WHICH THE PLT INITIALLY RESPONDED 'YES' AND A FEW SECONDS LATER ANOTHER VOICE FROM WHAT I BELIEVE TO BE THE SAME AIRPLANE SAID 'NO; WE'RE GOING AROUND.' I THINK THE SECOND RESPONSE FROM THE APCHING ACFT WAS MISSED BY MY CAPT BECAUSE AT THIS POINT HE WAS MAKING OUR STANDARD COMPANY CALLOUTS AND ENG INSTS CHKS; BUT IF HE DID HEAR THAT CALL FROM THE ACFT HE MISSED THE NEXT CALL FROM TWR TELLING US TO 'CANCEL TKOF CLRNC.' I THINK BY 100+ KTS I HAD COMMUNICATED THAT CALL FROM TWR TO MY CAPT AND HE DECIDED THAT THE SAFEST COURSE OF ACTION WAS TO CONTINUE THE TKOF AS WE HAD ENTERED INTO THE HIGH-SPD RTO SPD RANGE. IT SEEMED AT THE TIME THAT WE WERE NOT ENTIRELY CLR WHETHER THE OTHER ACFT WAS DEFINITELY GOING AROUND. AS I RECALL THE TCASII SHOWED A SOLID AMBER CIRCLE WITH A +03 AND A DOWNWARD FACING ARROW NEXT TO IT. THIS TARGET WAS JUST BEHIND THE AIRPLANE SYMBOL ON OUR MOVING MAP DISPLAY ON EFIS. THEREFORE; IN THE FEW SECONDS WE HAD TO MAKE THE GO/NO-GO DECISION IT APPEARED MOST REASONABLE TO ACCELERATE; CONTINUE THE TKOF; MAKE THE TURN TO 035 DEGS AS INITIALLY CLRED; MONITOR THE TCASII AND PUT AS MUCH DISTANCE BTWN US AND THE APCHING ACFT AS QUICKLY AS POSSIBLE. BY THE TIME WE LIFTED OFF; THE TCASII SEEMED TO INDICATE THAT THE REGIONAL JET HAD MADE A CLBING R TURN AWAY FROM US AND THE 035 DEG HDG WE TURNED TO WAS TO THE L. I THINK ONCE WE WERE AIRBORNE THE CAPT TOLD TWR THAT WE COULD NOT HAVE ABORTED AND THE CTLR HANDED US OFF TO DEP CTL. THOUGH TCASII DISPLAYED THE SOLID AMBER CIRCLE DEPICTING THE APCHING ACFT; OUR LOW ALT PRECLUDED IT FROM MAKING ANY AURAL WARNINGS OR RA'S; BUT EVEN SO; IT WAS DEFINITELY A GREAT TOOL TO HAVE.