The phrase to cast nasturtiums on, or at, is a deliberate malapropism punning on to cast aspersions on—cf. also I resemble that remark.

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The noun nasturtium denotes any of various plants of the genus Tropaeolum (family Tropaeolaceae) having round leaves and yellow, red or orange trumpet-shaped spurred flowers.

Originally, nasturtium denoted any of various plants of the family Brassicaceae (Cruciferae) having a pungent taste, especially garden cress and watercress.

This word is a borrowing from classical Latin nasturtium, which, in antiquity, was interpreted as being from nāsus, the nose, and tortus, meaning twisted, from torquēre, to twist, to distort, because the pungent smell of the plant causes one to wrinkle one’s nose. For example, the Roman statesman and scholar Pliny the Elder (23-79) wrote the following in The Natural History (Naturalis Historia – 77), a vast encyclopaedia of the natural and human worlds:

nasturtium nomen accepit a narium tormento
     translation:
Cress has got its Latin name from the pain that it gives to the nostrils.

And, according to Isidore of Seville (Isidorus Hispalensis – circa 560-636), Spanish archbishop and doctor of the Church, in his encyclopaedic work Etymologiarum sive Originum Isidori Hispalensis:

nasturtium sapor appellauit, quod acrimonia sui nasum torqueat
     translation:
Cress is named for its savour, because its bitterness wrenches one’s nose.

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These are the earliest occurrences of to cast nasturtiums on, or at, that I have found, presented in chronological order:

1-: From an extract from a comic paper titled Judy (i.e., probably, Judy, or the London Serio-Comic Journal), published in several British newspapers in October 1902—for example in The Shepton Mallet Journal (Shepton Mallet, Somerset, England) of Friday the 17^th:

“People have no right to go about casting nasturtiums on other people,” indignantly declared the old lady who was paying Mrs. Judy a visit the other afternoon.

2-: From The Strangers’ Wedding: The Comedy of a Romantic (Boston: Little, Brown and Company, 1916), by the English author Walter Lionel George (1882-1926)—Huncote has arrived at a ball:

Several of the young men glanced at Huncote, then ostentatiously turned away. They were jolly and chaffing as no girls were there to make them self-conscious; he heard an argument as to whether one of them preferred ginger beer. There seemed to be a lot of guffawing over this; then somebody in a very tight blue suit and a very high collar, obviously a clerk, said: “All right, ’Erb, I don’t want to cast any nasturtiums on you!”
Huncote pondered over this for some time.

3-: From “In England Now!”: A Weekly Letter from “Blanche”, published in The Bystander (London, England) of Wednesday 22^nd January 1919:

A gunner man writes to me from Germany: “Has the real cause of the Hun carrying the war into other people’s countries ever struck you, Blanche? It never did me until we marched through Cologne, en route for the other side of the Rhine via the Hohenzollern Bridge. Not once, during the whole hour that it took us to march through the city, did I see a pretty woman. Not that I’m casting nasturtiums at the sex, for the men are no better. And we have to live here for X months! Or is it years?”

Illustration by Helen McKie for “In England Now!”: A Weekly Letter from “Blanche”—The Bystander—22^nd January 1919:

“Not once . . . did I see a pretty woman”

4-: From If I become Clerk of the Weather. Things that will make some people wake up!, by A. P. Garland, published in The Sunday Post (Glasgow, Lanarkshire, Scotland) of Sunday 11^th April 1920—Clerk of the Weather denotes an imaginary functionary humorously supposed to control the state of the weather:

Another lot of bipeds with whom I have a sort of account to settle is the no-hat brigade, who drove me from a certain garden city. Because I did not choose to go forth in a strong sun with my brow bared they called me names. They cast nasturtiums upon the perfectly good hat I wore.