Did Sauron only want Middle Earth?

[Tuor in Gondolin]

Originally posted by Kamexkoopa
Middle-Earth is mostly the European part of the world

More than "mostly" the European part of the world. For example, in
Letters #165:

Middle-earth...is just a use of Middle English middel-erde
or (erthe), altered from Old English Middengeard: the
name for the inhabited lands of Men 'between the seas'. And although I have
not attempted to relate the shape of the mountains and land-masses to what
geologists may say or surmise about the nearer past, imaginatively this
'history' is supposed to take place in a period of the actual Old World of
this planet.

One way of giving depth to Middle-earth is to only
describe in detail a small portion of it (what is, in effect, Europe) and
describing only vaguely, as someone in much of the medieval period, would
have known of Asia and most of Africa, and probably view it as menacing and mysterious.

Hence, Middle-earth is conceived as Europe, Africa, and Asia, in a time when
(Tolkien would teasingly now allude to continental drift as cause) the Old World
land mass "between the seas" of the Atlantic and Pacific were different.

[Zigûr]

I believe it's true to say, is it not, that "Middle-earth" as a continent included the vast lands of Rhûn and Harad as well as "the West", ie Gondor, Rohan, Rhovanion and Eriador? At the end of the Third Age Sauron already more or less controlled all of the East and the South which seemingly made up the majority of Middle-earth's landmass.
It might be worth noting this remark from letter 183:

Sauron desired to be a God-King, and was held to be this by his servants; if he had been victorious he would have demanded divine honour from all rational creatures and absolute temporal power over the whole world.

As well as this remark about Sauron from Morgoth's Ring:

He probably deluded himself with the notion that the Valar (including Melkor) having failed, Eru had simply abandoned Eä, or at any rate Arda, and would not concern himself with it any more. It would appear that he interpreted the "change of the world" at the Downfall of Númenor, when Aman was removed from the physical world, in this sense: Valar (and Elves) were removed from effective control, and Men under God's curse and wrath.

What this would suggest to me is that Sauron's policy in the Third Age was the domination of all of Middle-earth but just Middle-earth; that way he would be the master of as much of Arda as possible while still apparently being free of the interference of the Valar or Eru (although of course he was lying to himself when he thought that either party had abandoned Middle-earth). Sauron knew, or at least thought he knew, his limits, it would seem, and the most powerful he could be was as ruler of the mortal lands of Middle-earth.
So I would say the answer is this: Sauron did only want Middle-earth, but Middle-earth was more than just the West. That being said, he already controlled the rest of it. Only the West was holding out.

[William Cloud Hicklin]

Remember, in the Third Age Aman and Eressea were no longer part of "the Earth;" Tolkien implies at the end of Akallabeth that the proto-Americas were created at the time of the Downfall and the World Made Round. I have no doubt that ultimately Sauron would have liked to control the New (literally) World as well.

[Celuien]

I don't think Sauron would ever have been satisfied with partial dominion. Greed and desire never seem to be satisfied in the beings who have gone to the evil side. Ungoliant's never ending hunger comes to mind, along with Morgoth's dissatisfaction with staying in his realms in the North (although, granted his original desire was for all of Arda). I'd guess that Sauron would fall into the same mold.

[Rhod the Red]

"This time spent east could've been part of Saruman's corruption and betrayal"

That's total rubbish. It was prompted by his study of Ring-lore and Souron's black arts.

You can't rewrite a storyline, esp. after publication.

[Legolas]

Total rubbish? I say "could've" and "part," and I stand by them.

He spent time out east where Sauron and Morgoth had influence dating back to the First Age. There's nothing contradictory about suggesting he could've picked up certain arts - even specifically some of Sauron's arts, as you say - or just mentally came closer to succumbing to his own desire of power.

He was gone a thousand years. Don't think he was playing Bingo that long.

[Boromir88]

Originally Posted by Legolas

He spent time out east where Sauron and Morgoth had influence dating back to the First Age. There's nothing contradictory about suggesting he could've picked up certain arts - even specifically some of Sauron's arts, as you say - or just mentally came closer to succumbing to his own desire of power.

I agree. Saruman is perhaps my favorite villain created in literature, such a fascinating lad.

Anyway, his personality is someone who loves to study things and absorb knowledge. Even if he wasn't evil at the time he went East, his personality (like Boromir's) had little defects that if unchecked would definitely lead down a path to evil. He was proud (led to lusting for power to the point that he was blinded by it and I would say even paranoid about losing his authority) and jealous of Gandalf, even back in Valinor, but especially after figuring out Gandalf had been given one of the elven rings:

And the Grey Messenger took the the Ring, and kept it ever secret; yet the White Messenger (who was skilled to uncover all secrets) after a time became aware of this gift, and begrudged it, and it was the beginning of the hidden ill-will that he bore to the Grey, which afterwards became manifest~Unfinished Tales; The Istari

And taking into account what Treebeard knows of Saruman:

"...he has a mind of metal and wheels; and he does not care for growing things, except as far as they serve him for the moment."~Treebeard

"I used to talk to him. There was a time when he was always walking about my woods. He was polite in those days, alway asking my leave (at least when he met me); and always eager to listen. I told him many things that he would never have found out himself; but he never repaid me in like kind."~ibid

The sentenced I bolded reveals a lot about Saruman (even in days when he was polite and "gave no trouble to his neighbours" - as Treebeard put it). He was a leech, a leech for knowledge but never "repaying" what he had learned. If it was of interest, and useful to Saruman he would take it, but he was not the giving type.

So, travelling East with the Blue Wizards I would say that yes, Saruman would have absorbed whatever knowledge he had a mind to take. We know that it was his devices which ultimate drove Sauron out of Dol Guldur too. This thirst for taking knowledge, yet not sharing any, probably contributed to his treachery. So, I think his knowledge of Sauron's mind and schemes (not to mention the study of Ring-lore) was unrivalled, even by Gandalf. At least until the time when Saruman falls to his lust for power, he becomes blind to the fact he has become a generic brand of Sauron.