But would the Irish bring in women and start a thriving population there? I think if it's not settled by the Norse, eventually word would get out about the Irish monks, the fishing grounds, and unused pastures on the island and decided to go over and settle it.
The Norse discovery and settlement of Labrador / Terra Nova seems to be almost an inconsequential event. However, a no-Nordic or Danish Greenland could have major geopolitical effects on North Atlantic. Could the island be a flashpoint to a U.S / British-Canadian dispute for the land?
Butterflies aside, the US has no capability to claim Greenland before the late 19th century if they're attempting to claim it from the British Empire. And by the late 19th century they have no little reason to claim it either anymore than they have reason to claim Newfoundland or Jamaica. It's not worth starting a war for, although they'd probably buy it if offered.
Before the 19th century Greenland is probably uninhabited by everyone except Inuit, missionaries, and maybe a Royal Navy outpost so is pretty inconsequential. It would likely have seasonal fishing and whaling camps.
There were Irish monks there when the Vikings showed up, and of course, they killed them. So maybe some Irish colonize it and then some Inuit from Greenland move in and you get the two cultures meeting and maybe clashing.
The OTL Inuit are unlikely to settle Iceland because they weren't well-established on the harsher east coast of Greenland and they'd have to cross a lot of sea ice, including open sea, to get to Iceland.