MULTIPLE RWY OP INTERSECTING RWYS ACR X HAD NMAC WITH ACR Y.

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ACR X WAS ISSUED A TKOF CLRNC ON RWY 31. OTHER ACFT WERE LNDG ON RWY 22. AS ACR X APCHED V1 WE OBSERVED ACR Y ON SHORT FINAL RWY 22. HE QUESTIONED HIS LNDG CLRNC AND THE TWR IMMEDIATELY ISSUED A GAR CLRNC. ACR Y EXECUTED A PULL UP WITH THE LOWEST POINT APPROX 50 FT AGL. I DELAYED ROTATION UNTIL WE WERE THROUGH THE RWY 31/22 INTXN. ACR Y PASSED DIRECTLY OVERHEAD AT WHAT I ESTIMATE TO HAVE BEEN 50 FT CLRNC WITH OUR TAIL. WE LIFTED OFF AT THE END OF RWY 31 AT A SPD WELL ABOVE V2. INTERSECTING RWY OPS ARE NORMAL SO EVEN THOUGH OUR TKOF CLRNC WAS NOT ISSUED UNTIL AN MLG WAS WELL THROUGH THE RWY 31/22 INTXN; I ASSUMED THERE MUST STILL BE SPACING IN THE GAP BTWN ARRS. OUR TKOF ROLL WAS EXECUTED IN A TIMELY MANNER; BUT IT LOOKED AS IF THE CTLR JUST LOST TRACK OF ACR Y. WHEN I SAW ACR Y; HE APPEARED TO BE HIGH ENOUGH SO THAT IF HE CAME NO LOWER; WE WOULD BE ABLE TO PASS BENEATH HIM. OUR SPD WAS SUCH THAT HAD I ATTEMPTED A REJECTED TKOF; WE MAY NOT HAVE BEEN ABLE TO STOP WITHOUT GOING OFF THE END OF THE RWY. HAD WE NOT STAYED ON THE RWY AS WE PASSED BELOW THE THEM I AM SURE WE WOULD HAVE HAD OUR TAIL CLIPPED. THE DAY WAS A HAZY BRIGHT OVCST DAY AND THE OTHER LGT WAS PAINTED PLAIN WHITE AND THAT MAY HAVE MADE IT HARDER TO SEE HIM EARLIER. SUPPLEMENTAL INFO FROM ACN 210516: ACR Y (DCA-LGA) WAS CLRED TO LAND ON RWY 22 AT LGA. BTWN 200 FT AND 400 FT; THE TWR INSTRUCTED US TO GAR BECAUSE THE ACFT AHEAD OF US HAD NOT CLRED THE RWY. THE GAR WAS EXECUTED PROMPTLY WITHOUT A PROBLEM. WE WERE COMPLETELY UNAWARE OF THE ACR X IN QUESTION UNTIL THE NEWS MEDIA RPTED THE INCIDENT.