The controller didn't overstep his boundaries, and he didn't tell the pilot to ignore a weather warning.
Shamrock demanded something that's extreme for the NYC airspace: runway heading for 15 miles. Usually asking for right or left deviation is sufficient, and that appears to be what the previous Delta departure did.
Once Shamrock declined an easterly heading, the controller had no choice but to vector them in a temporary hold. The controller was frustrated because the previous departures had been accepting turns to the east and his radar didn't show signs of significant precipitation. He also astutely pointed out that this weather was 5 miles from the runway departure end. Aer Lingus should have been able to paint that on their radar or see it with their eyes before they departed. Their clearance was to the GREKI fix so they knew they'd be heading that way after departure. That doesn't mean Shamrock was wrong. After all, they are flying in actual, and the controller is sitting in a room at zero knots, but in the incredibly dense NYC airspace, it can create an emergency if you have to deviate that much.
The controller was harsh and made his frustration evident in the way he chose his words. The pilot stroked his ego by saying his boss would call the controller's boss.
Once you strip away the emotions, the pilot and controller did their jobs. The pilot kept his airplane clear of weather that he determined to be dangerous. The controller did the only thing he could to keep the Aer Lingus flight out of that weather and away from conflicting traffic.