Well, we already have our National Day to Celebrate Being Irish, that's Paddy's Day :) (I suspect you're Irish, yourself, but this is for general audiences)

As you observe, Ireland is perhaps a little unusual in that it has several 'stages' of independence, and the actual independence itself is perhaps least well known. Unlike any American kid who will say 4th July 1776, in Ireland there's the 1916 Rising (24 MAR 1916), the Declaration of Independence (21 JAN 1919), and then the Creation of the Irish Free State (06 Dec 1922), the Adoption of the Irish Constitution (thus creating Ireland, 29 Dec 1937) and finally the Creation of the Irish Republic, 18 April 1949. Believe it or not, I had to check the last date. I don't recall it ever even being mentioned in class, though maybe I was sick that day.

Of these, and possibly closest to the US Independence Day, we also have the day to celebrate fighting the British to get them out, that being Easter, given the Easter Rising started Easter Monday, 24 March 1916. Last year, the 100th Anniversary, it was Easter Sunday, March 27th, which saw the annual Reading of the Proclamation of the Republic, a large military parade through town, etc. The exact date of the ceremonial Reading varies by year, depending on the Easter calendar. There would be a military parade every year until 1970, when issues started cropping up with Sinn Fein. Then we had the issues up North, and discussion started as to whether or not it was even a good idea to celebrate the Easter Rising as it was something which was likely to inflame tensions, so things got a little quiet. Although the Reading of the Proclamation has been conducted by a military officer at the front of the GPO (General Post Office, ground zero for the Rising) every Easter Sunday, only the major anniversaries resulted in a large military parade since 1970, notably 2006 and 2016.

Oddly (Unless it's the trigger for your question), it happens that by coincidence there's currently a bill going through the Dail (that's parliament) to declare a Declaration of Independence Day of 21 January. It wouldn't be an official day off, or anything, more a commemorative thing based on the basis that 21 January was the date of the meeting of the first Dail, which was also when the Irish Declaration of Independence was made (which then kicked off the War of Independence).