We need to remember that U.S. population was increasing not only thanks to immigration, but also thanks to its high natural growth. I've just made a calculation trying to establish how big share of U.S. population in 1980 was descended from the "old stock" of 1790.
I've found data on Rate of Natural Increase (livebirths minus deaths) per 1000 population per year in this publication:
http://www.nber.org/papers/h0056
Here is the data (if I counted it correctly, avg. RNI per 1000 pop. per year in period 1790-1980 was 17,05 - so 1,705 %):
http://www.nber.org/papers/h0056.pdf
The population of the USA by the time of the 1st census - in 1790 - was 3,929,625 including 3,172,444 White people.
So let's apply annual natural growth of 1,705 % to 3,172,444 people in period 1790 - 1980, using this calculator:
http://www.metamorphosisalpha.com/ias/population.php
The result is 78,782,861:
So had there been absolutely no immigration in period 1790 - 1980, White people in the USA would have numbered ca. 79 million in 1980.
Of course this is all under the assumption, that those of the "original 1790 stock" had avg. RNI rate of 1,705% annually in period 1790-1980. This assumption doesn't necessarily has to be correct. They could have a slightly different avg. RNI rate just as well.
Now acccording to the same study (wikipedia also confirms), in 1980 the USA had 226,545,805 inhabitants.
Of them only 180,256,103 were "Non-Hispanic Whites" (and 669,799 were "Others", probably including mixed-race Whites):
http://www.censusscope.org/us/chart_race.html
Another source says that Non-Hispanic Whites numbered in total 180,603,000 people in 1980 (see Table 1 on page 11):
http://www.prb.org/Source/54.3AmerRacialEthnicMinor.pdf?q=543-minorities
Let's round this number up to 181 million and we can say, that some 78 million of them (ca. 43%) were descendants - mostly (because of course all of them were probably mixed with later immigrants) - of White population of the "original 1790 stock".
But who were those people of "original 1790 stock"?
I've already presented this data on various pages of this thread before, but let's post them again:
I've found two different estimates of ethnic groups:
I. First estimate is from:
"A Century of Population Growth: From the 1st Census of the U.S. to the 12th 1790-1900", 1909:
Black African - 19,27%
Whites - 80,73% - including:
English & Welsh - 66,31%
Scottish - 5,64%
Irish - 1,57%
============
British-Irish total - 73,52%
============
German - 4,47%
Dutch - 2,01%
French - 0,45%
all other whites - 0,29%
Almost identical data in: S. P. Orth, "Our Foreigners: A Chronicle of Americans in the Making".
II. Second estimate is from:
"United States Ethnic Groups in 1790: Given Names as Suggestions of Ethnic Identity", 1989:
Black African - 19,0%
Whites - 81,00% - including:
English - 48,00%
Welsh - 3,50%
Scottish & Scotch-Irish - 12,80%
Irish - 4,70%
============
British-Irish total - 69,00%
============
German - 7,20%
Dutch - 2,50%
French - 1,70%
Jewish - 0,25%
Swedish - 0,20%
other whites - 0,15%
Applying a separate count in the calculator for each decade, with RNI for each decade separately, takes much more time but gives a very similar (slightly lower) final result - 78,5 million (instead of 78,8) in 1980. Descendants of the "1790 stock" (increase by natural growth in each decade):
1790 - 3,172,444
1800 - 4,120,417
1810 - 5,370,457
1820 - 6,854,544
1830 - 8,941,018
1840 - 11,297,613
1850 - 14,165,554
1860 - 17,327,074
1870 - 20,637,967
1880 - 24,547,716
1890 - 28,682,886
1900 - 32,566,719
1910 - 36,994,705
1920 - 41,455,652
1930 - 46,855,645
1940 - 50,345,679
1950 - 56,774,529
1960 - 66,110,204
1970 - 73,541,777
1980 - 78,456,469
I made a graph showing hypothetical U.S. population growth until 1980 had there been absolutely no immigration since 1790:
http://s28.postimg.org/m3mus5j3v/USA_Pop_Growth.png
One might also observe that the Irish tend to be particularly proud of their heritage, and suggest a link.
In year 1900 out of the total of 76 million inhabitants of the USA, 1.6 million were Irish-born, another 5 million were born to Irish-born parents (i.e. those were 2nd generation Irish immigrants) and only God knows how many were 3rd and 4th generation Irish immigrants. So I would rather trust the census when it says that 30 million people reported Irish ancestry in 1980 (when the USA had 226 million people), given the high rate of natural increase of American population between 1900 and 1980, as well as the rate of intermarriage of Irishers with others (i.e. the dillution, but also the faster spread, of Irish ancestry).
After year 1845 there was simply a long-lasting mass exodus of Irish population from Ireland to other places, mostly to the USA.