B737-800 ON LNDG ROLLOUT AND CARJ CLEARED TO CROSS AT TAXIWAY F EXPERIENCE NEAR COLLISION ON RWY 22 AT LGA.

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WE FLEW AN ILS APCH TO RWY 22 AT LGA WITH FLAPS 30 DEGS; VREF OF ABOUT 135 KIAS AND MANUAL BRAKES. WE WERE CLRED TO LAND BY LAGUARDIA TWR. WHEN WE WERE STARTING THE FLARE A REGIONAL JET LOOKED LIKE IT WAS STARTING TO MOVE L TO R TO CROSS RWY 22 ON TXWY E. FO CALLED THIS OUT. AS WE TOUCHED DOWN WE REALIZED THAT THE REGIONAL JET WAS NOT GOING TO STOP SHORT AND CONTINUED TO CROSS RWY 22 IN FRONT OF US. FO CALLED BRAKES AND WE BOTH IMMEDIATELY APPLIED MAX MANUAL BRAKING AND I APPLIED FULL REVERSE THRUST. WE SLOWED VERY QUICKLY REACHING TAXI SPD PRIOR TO TXWY E. ABOUT THE TIME ACFT Y WAS CLRING THE RWY LAGUARDIA TWR MADE A RADIO CALL SAYING THAT THERE WAS AN ACFT EXITING AT MIDFIELD. WE CLRED THE RWY AT TXWY D AND TAXIED TO GATE. WHILE TAXIING TO THE RAMP WE ACCOMPLISHED THE AFTER LNDG CHKLIST. WHILE STOPPED ON TXWY A FOR TAXIING TFC WE REFED THE BRAKE COOLING FOLLOWING LNDG CHART IN THE ODM. WE SAW THAT WE WERE NOT IN THE CAUTION OR FUSE PLUG ZONE SO WE CONTINUED TO THE RAMP WHEN CLRED IN. AFTER BLOCKING IN WE ACCOMPLISHED THE SHUTDOWN CHKLIST. WE THEN REFED THE ODM TO COME UP WITH A MORE REFINED BRAKE COOLING TIME OF ABOUT 24 MINS; WELL LESS THAN THE TURN TIME OF THE ACFT. AS A PRECAUTION MAINT INSPECTED OUR BRAKES AND LNDG GEAR AND ALL WAS NORMAL. SUPPLEMENTAL INFO FROM ACN 744729: ON TAXI OUT WE WERE INSTRUCTED TO HOLD SHORT OF RWY 4 IN LGA ON TXWY F. WE COMPLIED AND HELD SHORT. AS WE WERE #1 FOR XING THE RWY SOMEBODY ADVISED GND THAT THERE WAS A VEHICLE ON FIRE ON THE OTHER SIDE OF THE ARPT NEAR THE 5 TWRS PARKING AREA. THE CTLR SAID HE WAS AWARE THAT THERE WAS A FIRE. SHORTLY THEREAFTER HE CLRED US ACROSS THE RWY AT TXWY F. I READ BACK THE XING CLRNC AND GLANCED OUT THE R SIDE OF THE ACFT AND OBSERVED NO TFC. IT IS NOTEWORTHY THAT TXWY F IS A HIGH SPD TXWY AND WHEN UTILIZED THE ACFT FACES TOWARDS THE OPPOSITE SIDE OF THE RWY AT A 45 DEG ANGLE. THIS MAKES IT VERY DIFFICULT TO SEE DOWN THE ENTIRE LENGTH OF THE RWY OR SEE THE FINAL APCH END. AS WE STARTED MOVING WE WERE TOLD TO EXPEDITE WHICH WE DID AND I RESPONDED. AS WE ENTERED THE RWY AND THE ACFT WAS PERPENDICULAR TO THE RWY I TOOK ANOTHER LOOK AND SAW A B737 ON THE RWY APCHING US RAPIDLY AND HE APPEARED TO BE USING MAX BRAKES. I TOLD THE CAPT THAT THERE WAS AN ACFT APCHING AND TO MOVE IT. HE COMPLIED IMMEDIATELY BY ADDING FULL THRUST TO THE ENG. WE CLRED WITH MAYBE 400 FT TO SPARE. I ALSO NOTICED THERE WAS INDEED A DASH 8 ON A 1 MI FINAL; I THINK THE GND CTLR MUST HAVE LOST TRACK OF THE B737; POSSIBLY BECAUSE OF THE FIRE SITUATION. POSSIBLE WAYS TO AVOID THIS INCURRENCE ARE AS FOLLOWS: 1) WE SHOULD HAVE NEVER BEEN ISSUED THE XING. 2) CTLR WAS DISTR AND SHOULD HAVE SLOWED DOWN INSTEAD OF TRYING TO RUSH US ACROSS. 3) TXWY F OR ANY OTHER REVERSE HIGH SPD SHOULD NOT BE UTILIZED FOR XING AS IT IS VERY DIFFICULT FOR THE FLT CREW TO SEE DOWN THE RWY. 4) RWY XINGS SHOULD BE MADE WITH THE TWR CTLR AS HE OR SHE KNOWS FIRST HAND THE STATUS OF THEIR RWY.