Of Australian-English origin, the phrase don’t argue designates the straight-arm fend-off in rugby football—the gesture made by the player carrying the ball in this photograph, published in The Sydney Morning Herald (Sydney, New South Wales) of Tuesday 9^th June 1992:

In A Dictionary of Australian Colloquialisms (Sydney University Press in association with Oxford University Press Australia, 1990), Gerald Alfred Wilkes (1927-2020) defined the phrase and quoted a text explaining its origin:

don’t argue The straight-arm warding off movement in Rugby football [f. illustration in the trademark of J. C. Hutton Pty Ltd *: see quot. 1977]
[…]
1977 Australian 14 Apr. 18: Rugby football fans around the world know the straight-arm fend-off as a ‘Don’t argue’, and its origins go back to the turn of the century. The creator of the symbol was Mel B. Spurr, a pianist, singer, dancer, vaudevillian, monologist, cartoonist and story-teller. Mel Spurr starred in Melbourne’s Tivoli, the Athenaeum Hall and the Town Hall in the golden days of vaudeville. Spurr invented the two-man symbol and took it to Hutton’s Melbourne manager. The ‘Don’t argue’ slogan was quickly evolved, and in a short time became one of Australia’s best-known trademarks.

(* J. C. Hutton Pty Ltd is a smallgoods and curing manufacturing company established by James Carruthers Hutton in Melbourne in 1873—source: The Dictionary of Sydney.)

This advertisement for Hutton was published in The Bowen Independent and Proserpine Agriculturist (Bowen, Queensland) of Saturday 11^th March 1911. It depicts a man putting a hand in another man’s face, and it reads:

“Don’t argue” Hutton’s bacon is the best
“Nothing to argue about!”
Pineapple Brand hams bacon

All the earliest occurrences of the phrase that I have found are from accounts of rugby matches, published:

1-: In The Cumberland Argus and Fruitgrowers’ Advocate (Parramatta, New South Wales) of Saturday 6^th June 1914:

Hill secured the ball in the second half, and cut across to the wing. A Balmain back moved across for a tackle, but Hill gave him the “don’t argue” on the chin, and the Balmain player landed on his back.
The incident was very amusing.

2-: In The Lithgow Mercury (Lithgow, New South Wales) of Wednesday 10^th June 1914:

A Newtown man on one occasion tackled a man in the “don’t argue” fashion. Several others emulated his example, and some high feeling crept into the game, but it was only of short duration.

3-: In The Daily Standard (Brisbane, Queensland) of Monday 15^th June 1914:

Wagstaffe proved himself a wonderful player quick, strong, and safe, with a powerful hand which he used to fend off his opponents in the “don’t argue” attitude.

4-: In the Sydney Sportsman (Sydney, New South Wales):

4.1-: Of Wednesday 28^th June 1916:

Caples, who put up the best game since he came into first grade, was the best scorer for Easts, and a beautiful try it was. From a scrum about twenty yards out Male gave him a sharp hard pass and with his don’t argue fend, which he was so noted for at St. Joseph College, he kept all tacklers off, and ran right in near the posts, and Messenger easily goaled.

4.2-: Of Wednesday 26^th July 1916:

Caples gave a splendid exhibition of the “don’t argue” fend. First he skittled Ricketty Johnson, and the pride of Oatley saw stars and comets for a few seconds; two forwards were treated in the same way.

4.3-: Of Wednesday 9^th July 1919:

There is no doubt that Harry Caples has the best “don’t argue” fend in Sydney at the present time.

5-: In The Rugby League News (Sydney: New South Wales Rugby Football League) of 6^th June 1921:

The most surprised man on the ground when Watt scored Balmain’s first try was Gordon Wright. The latter attempted to smother the Balmain hooker as he was endeavouring to improve the position for the kick at goal, but Watt’s “don’t argue” fend made Gordon sit down suddenly with a pained expression on his face.

I have found verbal uses of don’t argue in the account of a rugby match, published in the Darling Downs Gazette (Toowoomba, Queensland) of Saturday 12^th June 1920:

Gordon […] has even a superior fend-off than Nev. Broadfoot, as when “don’t arguing” he moves his feet away at the same time, which makes him practically immune from even a diving tackle.
[…] The only bloomers “Nigger” made during the game were when Gordon “don’t-argued” him on two occasions.

I have found two cartoons parodying the advertisements for Hutton.

1-: This cartoon was published in The Mercury (Hobart, Tasmania) of Thursday 25^th August 1938. England had won the Test match at Kennington Oval on Monday 24^th August 1938, inflicting on Australia the heaviest defeat in the history of Test cricket (one of the English players was called Hutton). The cartoon depicts an Englishman putting a hand in the face of an Australian man, and saying “Don’t argue—Hutton is the best!”

2-: This cartoon by Nicholson is from Whatever happened to ’76?, published in The Age (Melbourne, Victoria) of Saturday 1^st January 1977. It depicts a man saying “Don’t argue! Devaluation is best!” while putting a hand in the face of another man, who says “It’s moments like these you need cyanide”:

This cartoon illustrated the following text:

There were many highlights (or were some of them lowlights?) in the utterances and attitudes of the famous and not-so-famous in 1976. These are a few:
Devalue? Never! What, never? Well, hardly ever.
Who said it first? Fraser? Lynch? Even Anthony . . . no, couldn’t have been him. It might even have been the Governor of the Reserve Bank, the Secretary of the Treasury or perhaps the Commissionaire of the Royal Mint.
Whoever it was, it was the Quoting Man’s Quote: “We will not devalue.” (Remember, only 360 devaluation days until Christmas.)