In Milton’s Paradise Lost, book II (1667), Belial counsels against an assault on Heaven:

What if the breath that kindl’d those grim fires
Awak’d should blow them into sevenfold rage
And plunge us in the flames? or from above
Should intermitted vengeance arm again
His red right hand to plague us?

This last image is lifted from Horace’s Odes I.2:

Iam satis terris nivis atque dirae
grandinis misit Pater et rubente
dextera sacras iaculatus arces
terruit urbem […]

Enough already of dire snow and hail
has the Father sent upon the earth,
and smiting with his red right hand
the sacred hill-tops has filled with fear the city […]

Interestingly, Horace’s “red right arm” image was, by not just one but two nineteenth-century poets, transposed from I.2 to Odes III.3 —

Iustum et tenacem propositi virum
non civium ardor prava iubentium,
  non vultus instantis tyranni
    mente quatit solida neque Auster,
dux inquieti turbidus Hadriae,
nec fulminantis magna manus Iovis
  si fractus inlabatur orbis,
    impavidum ferient ruinae.

For the above Latin, Byron gives us:

The man of firm and noble soul
No factious clamours can control;
No threatening tyrant’s darkling brow
Can swerve him from his just intent:
Gales the warring waves which plough,
By Auster on the billows spent,
To curb the Adriatic main,
Would awe his fix’d determined mind in vain.

Ay, and the red right arm of Jove,
Hurtling his lightnings from above,
With all his terrors then unfurl’d,
He would unmoved, unawed behold:
The flames of an expiring world,
Again in crashing chaos roll’d,
In vast promiscuous ruin hurl’d,
Might light his glorious funeral pile:
Still dauntless midst the wreck of earth he’d smile.

And Tennyson (c. 1823) — whose idol Byron died, aged 36, when Tennyson was only 14 himself) has:

The people’s fury cannot move
The man of just and steadfast soul
  For he can brook
  The tyrant’s look
And red right-arm of mighty Jove […]

Hat tip to the quite interesting journal article “Barbarous Hexameters and Dainty Meters: Tennyson’s Uses of Classical Versification” (A. A. Markley, Studies in Philology, 1998).

From Horace and Milton (I guess?) “the red right arm of Jove” becomes a stock image. Alexander Pope somewhat anachronistically inserts it into his (1726) translation of Homer’s Odyssey XXIV:

With dreadful shouts Ulysses pour’d along,
Swift as an eagle, as an eagle strong.
But Jove’s red arm the burning thunder aims:
Before Minerva shot the livid flames;
Blazing they fell, and at her feet expired […]

And when Joseph Addison, in his Latin “Battle of the Pygmies and Cranes” (c. 1690), writes merely “flagrantia tela deorsum torquentur Jovis acta manu,” Thomas Newcomb’s translation (1724) inserts “the dreadful red right-hand of Jove” and William Warburton (also 1724) says “Jove’s red arm.”

William Julius Mickle inserts “Jove’s red arm” into his (c. 1776) translation of Camões’ Lusiad V.58.

Meanwhile, TVTropes went with the phrase as the “trope namer” for something else entirely — the trope of the abnormally-handed villain (Captain Hook, Darth Vader, Count Rugen), or perhaps any evil deformity in general. I’m not sure how they settled on Horace’s phrase to represent that trope!

Serendipitously in light of Milton’s 1667 date, it seems that the first observations of a great red spot on the planet Jupiter are generally dated to 1664 (Robert Hooke) and 1665 (Giovanni Cassini). However, while both Hooke and Cassini recorded some kind of spot on Jupiter, neither one specifically recorded it as red; and in fact Hooke’s “spot” was more likely just the shadow of a transiting Callisto.