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Irish folks often have a surname that begins with the prefix "Mc."
Irish immigrants in the early United States became commonly known as "Micks."
The Irish are predominantly Catholic. The rest of the American population was and still is mostly Protestant.
Because of this, the Irish often sent their children to separate Catholic schools, and they attended Catholic churches.
In the early US, these Catholic institutions became synonymous with the Irish. Everyone who went there was pretty much assumed to be Irish.
To outsiders, everyone at a Catholic school or church was a "Mick."
The other notable group that was started coming to these schools and churches was Spanish immigrants. They went everywhere that the Irish did.
So people called them "Spanish Micks," which was shortened to "Spics."