Doing a google search for "pretty little Irish girl" took me to some very odd sites, but if it didn't turn up the song Mary is looking for it did suggest some possible leads.

There was a 1926 Daryl F Zanuck silent movie called 'The Little Irish Girl' starring Dolores Costello in the title role luring the innocent John Harron into a gambling den. Could this be based on an earlier Broadway show?

Or there is a song composed by Hermann Lohr and Edward Teschemacher in 1903 which Rufus Wainwright apparently sings.

The Little Irish Girl (Duet)

This version is set up for two voices.
Each verse has four lines, but due to the two
different parts, it may not look that way on paper.

M) As I went out one evening to Tipperary town,
M) I met a little colleen a-moung the heather brown.
M) Oh, says I perhaps you're lonely, she tossed her pretty curl.
F) Well maybe I prefer it!
M) Och, the dear little girl.

M) Says I, perhaps you're married?
F) Says I, perhaps I'm not!
M) Says I, I'll be your gossoon,
F) Says I, I'll not be caught!
M) Oh, your eyes are like the ocean, and your heart is like a pearl.
F) 'Tis true, then I will keep it!
M) Och, the dear little girl.

M) Says I, I've got a cabin and pigs that number seven,
M) And oh, with you ma-yourneen sure, the place would be like heaven.
F) Oh, I looked into his eyes then, my heart was in a whirl!
M) The little pigs had done it! Och, the dear little girl!
F) The little pigs had done it!

There's another song [or tune?] "I'm going to marry that pretty little girl" recorded by the Sweet Brothers which appears on County records LP 535 'Round the Heart of Old Galax Vol.3 - recordings 1924-37'.

Ruby Murray recorded a song called 'A Pretty Irish Girl' but I can't find the words to it.

To be honest Malcolm's reminder that the chorus of 'Step it out, Mary' comes from an Irish skipping rhyme has made me wonder whether the rest is related in some way to the children's game song 'Pretty Little Girl of Mine' although the words which Mary's father remembers are quite distinctive.

One factor that might help in tracking down this extremely elusive song is knowing whereabouts in his youth that Mary's father might have heard it.