The 'United Kingdom' is a sovereign state constituted by the four countries of England, Northern Ireland, Scotland and Wales, as set out in the Act of Union (1800), the first article of which states that
That Great Britain and Ireland shall upon Jan. 1, 1801, be united into one kingdom; and that the titles appertaining to the crown, &c. shall be such as his Majesty shall be pleased to appoint.1
Ireland was partitioned into 'Northern Ireland' and 'Southern Ireland' under the Government of Ireland Act (1920), but both remained a part of the United Kingdom:
On and after the appointed day there shall be established for Southern Ireland a Parliament to be called the Parliament of Southern Ireland consisting of His Majesty, the Senate of Southern Ireland, and the House of Commons of Southern Ireland, and there shall be established for Northern Ireland a Parliament to be called the Parliament of Northern Ireland consisting of His Majesty, the Senate of Northern Ireland, and the House of Commons of Northern Ireland.2
The parliament of Southern Ireland never took effect under this arrangement, and the territory instead seceded from the United Kingdom after the Anglo-Irish Treaty was implemented in 1922, taking on 'Dominion' status within the British Empire and becoming its own sovereign entity.1 Under the treaty, the Northern Irish had the right to opt-out of the new Irish Free State, but in the interim (a matter of days) between the enactment of the treaty and the formalisation of this opt out, it was technically a part of the Irish Free State and thus, for a very brief period, I suppose not a part of the United Kingdom (even on this technicality, though, your Irish friend still isn't right).
Since then, though, the United Kingdom has been thus constituted by Great Britain and Northern Ireland. It has not, since the 1800 Act of Union, ever had the seemingly contradictory status of being a part of the United Kingdom yet somehow a satellite to it, as your friend has suggested.
Other references (to rub his face in it):
The BBC's United Kingdom Country Profile:
The United Kingdom is made up of England, Wales, Scotland and Northern Ireland.
The OED's definition of the United Kingdom of Great Britain and Northern Ireland:
A country of western Europe consisting of England, Wales, Scotland, and Northern Ireland; population 61,113,200 (est. 2009); capital, London. Full name United Kingdom of Great Britain and Northern Ireland.
England (which had incorporated Wales in the 16th century) and Scotland have had the same monarch since 1603, when James VI of Scotland succeeded to the English crown as James I; the kingdoms were formally united by the Act of Union in 1707. An Act of Parliament joined Great Britain and Ireland in 1801, but the Irish Free State (later the Republic of Ireland) broke away in 1921. The UK became a member of the EC (now the EU) in 1973
In sum: he was wrong, you were right.