Lighter has posted a link to Robinson's singing of "Rio Grande," and links to other songs recorded by Robinson in thread 72642: Robinson Rio
Rather than muddy that thread, here is one for other versions of this old song and chantey.
Perhaps the first mention in print is by G. W. Sheldon, 1882, Harper's New Monthly Magazine, July, pp. 281-286, "Sailor Songs."

"The most pretentious, though not always the most meritorious, of windlass songs were those in which the second chorus was greatly extended, and made in some instances longer than all the rest of the song. Of these, there is one in which the chorus rises and swells with the crescendo of the heaving Atlantic swell" (music given).

RIO GRANDE
(Sheldon, 1882)
Solo:
I'll sing you a song of the fish of the sea.
Chorus:
Rolling Rio...
Solo:
I'll sing you a song of the fish of the sea.
Chorus:
To my Rolling Rio Grande.
Hurrah, you Rio,
Rolling Rio.
So fare you well, my bonny young girls,
For I'm bound for the Rio Grande.
No other verses given.

Speaking of the shanthman, Sheldon says: "If he was an artist of any real cultivation, he had at least seventy-five songs at his tongue's end."
Sheldon speaks disparagingly of the quality of the songs and mentions that many were "highly objectionable on the score of morality.." But hegoes on to say that they were no worse in this respect than the songs "which one occasionally hears in the smoking car of an excursion train..."
Already, in 1882, he said: "But both the good and the bad songs ceased when the sailor disappeared, and to revive them on the deck of an iron steam-ship would be as impossible as to bring back the Roman trireme."

RIO GRANDE
(Carmina Princetonia, 1894)

1. Where are you going, my pretty maid?
Heave away. Heigh ho!
I'm going a-milking, Sir, she said,
And I'm bound for the Rio Grande.

Chorus:
Heave away, heigh ho!
Heave away, heigh ho! heigh ho!
I'm going a milking, Sir, she said,
And I'm bound for the Rio Grande.

2. Oh, what is your father, my pretty maid?
Heave away, Heigh ho!
My father's a farmer, Sir, she said,
And I'm bound for the Rio Grande.

Chorus:
Heave away, heigh ho!
Heave away, heigh ho! heigh ho!
My father's a farmer, Sir, she said,
And I'm bound for the Rio Grande.

3. Oh! what is your fortune, my pretty maid?
Heave away, heigh ho!
My face is my fortune, Sir, she said,
And I'm bound for the Rio Grande.

Chorus:

4. Oh! then I'll not marry you, my pretty maid,
Heave away, heigh ho!
Oh! nobody asked you, Sir, she said,
And I'm bound for the Rio Grande.

Chorus:
Note- The Chorus repeats the last two lines of each verse.

Printed with music. M. Taylor Pyne et al., Editorial Committee, 1894, Carmina Princetonia, The University Song Book," 8th ed. Supplementary, Martin R. Dennis & Co., Newark, NJ.

Both of these versions are earlier than those mentioned in "The Traditional Ballad Index," but the songs seem to date back to the time when the coast and ports of Rio Grande do Sul were important to seamen. Stan Hugill discusses this in his books.