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My understanding is that "見やる" means " _to cast a casual glance at_." So, I'd
think that these sentences are synonymous?:

> 空を見やった。  
>  空を無頓着に見た。

"見すえる" is " _to intensely stare at something_ ". Thus, these are synonymous?:

> 速く近づいてきた虎を見すえた。  
>  速く近づいてきた虎を恐怖で見た。

Thus finally making "見やる" and "見すえる" antonyms? Any major nuance between the
two?
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2015-08-20T16:28:59.797
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word-choice nuances
Are ”見やる" and "見すえる" near perfect antonyms?
I don't know if 見やる and 見据える are antonyms, but anyway, unfortunately, many of
these sentences are not natural, and do not work as you expect.

> 空を見やった。

Yes, this means "gave the sky a glance".

> 空を無頓着に見た。

I don't think this sentence is natural because 無頓着 usually means "not to be
interested in something people should usually care for." (e.g., 彼は服装に無頓着だ = he
is indifferent to clothes). 頓着 means 'interest/heed'. Better ways to rephrase
the first sentence are 空を何気【なにげ】なく見た, 空に目をやった, etc.

> 速く近づいてきた虎を見すえた。

見据える (見る see + **据える settle** ) mainly takes intangible objects such as
夢/将来/現実, and means "to bear/keep in mind (for a long time)". It may also take
tangible/animate objects, but this word refers to seriously looking at
something with both time and care. I don't think that's what people normally
do against an approaching tiger.

> 速く近づいてきた虎を恐怖で見た。

This で in 恐怖で denotes the reason for the action. You can say 恐怖で震えた or
恐怖で動けなくなった, but I don't think 恐怖で見た is natural (to look due to fright??).
Instead, you can say 虎を恐怖に満ちた目で見た (to look at the tiger with frightened eyes).
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In addition to the good answer already given, here's an explanation of the
nuance of 見やる.

The action of 見やる can be a glance but does not have to be so.

The verb 見やる means " **see (someone/something) far away** " or " **look at
(someone/something) in the distance.** "

For example,

> 彼は虎を **見た** 。

this sentence simply tells "He looked at a tiger/tigress." We don't know the
distance between the place 彼 was at and the other place 虎 was at, unless
additional information is provided.

But if the verb in the sentence is 見やった instead of 見た, like this,

> 彼は虎を **見やった** 。

the sentence not only tells that 彼 looked at 虎, but also expresses that 彼 was
somewhere far from the 虎. How far away they are from each other is not
certain, but probably the distance is not short.

This is the nuance of the word 見やる. I don't think that 見やる and 見すえる are polar
opposites. They are just different.

I think that the meaning of 「無頓着に見る」 is almost equivalent to a verb 見[流]{なが}す.
But in general, it's strange to use 見流す when someone sees the whole sky. 見流す
is usually used to mention objects which have shapes. So, I guess that the
phrases which narutoさん has suggested fit the original meaning of your first
sentence more than this verb.

In terms of attention, 見流す has an opposite meaning of 見つめる／見[入]{い}る. But
technically, they are not polar opposites or antonyms, as far as I know.
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