Some of them did.
The name Scoti was originally a (4th century AD)Latin term for Gaels who raided Roman Britain from Ireland. By the 5th century these Gaels had formed a kingdom (Dalriada) which straddled the relatively narrow channel between north-eastern Ireland and south-western Scotland. However the country now known as Scotland was, in the Dark Ages, divided between these Gaels and several other races including Strathclyde Britons (Celts) forced north by the saxons(whilst other Celts remained in Wales and Cornwall), Norse moving south from Orkney and Shetland, Picts who occupied most of the central and northern highland areas and Angles moving north from Northumbria. A separate Viking group colonised Galloway. These disparate groups sometimes fought and sometimes didn’t.
Legend has it that in the mid-ninth century a Dalriadan king called Kenneth the Hardy was defeated in battle by the Picts, who were then attacked in the rear by a Viking army. The Vikings won, but as was their habit, then retired to the sea rather than holding the land. Kenneth’s army was now in better shape than the Picts and he was able to take over Pictish territory, becoming the first king of a united Alba (Dalriada + Pictland = just about all the mainland north of the highland line). Nowadays there is controversy about whether this amalgamation of the two kingdoms actually did occur by war or by dynastic marriage or some other process and not everyone considers Kenneth (MacAlpin) the first king of the unified land.
It took a lot longer for the lowlands (Strathclyde and the Lothians) to be incorporated into the kingdom and longer still for the Norse settlements in the isles of the west and north to be included too. Long before this process was complete it was complicated by the arrival of Normans who had already conquered England and who also invaded Ireland.
In the 17th century the colonisation process was reversed as protestants from Scotland were settled in Gaelic Ulster, which had been difficult to govern from the British side of the water.
The famine of the 1840’s set off more emigration from Ireland and significant numbers came to Scotland in a second wave of Irish settlement. Many got no further than Glasgow and surrounding west coast towns, most engaging in manual labour in the newly expanding industries.
So whilst numerous Scots today are of Irish descent, many others are descended from the various races that were amalgamated into the kingdom. These various strands are nowadays thoroughly mixed together anyway.