Short Answer: No.
This is a great question, and I'm glad you asked it. This verse is often used by Christian apologists to show that the Bible was ahead of its times in its scientific claims. While this sounds convincing to modern readers of English translations, it is a very poor argument to use.
Exhibit A: The word "stretch"
To many, the idea of God "stretching" out the heavens sounds an awful lot like the "expansion" of the universe, but that is not what the word means. The Hebrew word נָטָה has the idea of something being spread out. Imagine standing upright and sweeping your arm in an ark from one side to the other (like you were painting a rainbow) and saying "God spread out the heavens". This gives a good picture of what the word means. It can be translated (depending on context) as spread out, reach out, stretch out, extend, or bend.
Exhibit B: Context
The context really drives our understanding of Hebrew words and helps us move from semantic range to authorial intent. So what did the author intend to say here? Was he trying to explain an astronomical theory which wouldn't be discovered for another 3,000 years? No. Let's look at his own description of this "spreading out" of the heavens:
Who stretches out the heavens like a curtain
Do curtains experience a Big-Bang-esque ever-increasing expansion of internal matter? No. Curtains are simply spread out to cover a window. The author's meaning here is simply that God spread out the heavens like a covering.
Exhibit C: Hebrew parallelism
Any reader of the Hebrew text will notice that in poetic passages (exp. in songs, proverbs, & prophecy), parallelism abounds. The Hebrew people liked to say one thing with two words (or phrases.) They did this at times for clarity, at times for an appealing sound, and at times because... well... that's just the way they wrote! The two parallel words / concepts were often linked by "and". We have an example of that literary device here in this passage:
Who stretches out the heavens like a curtain
And spreads them out like a tent to dwell in.
The "stretching" is put in parallel with the "spreading"; these two are signifying the same thing.
Conclusion
The author's intention was simply to say that God spread out the heavens like a covering for the earth.
While in practice the Isa. 40:22 apologetic can be effective in reassuring believers and convincing non-believers that the Bible is the word of God, it should not be used, because that is not what the text is saying. We need to be careful to respect the authorial intent behind the words that we're reading and not use (abuse) the author's words for our own purposes -- particularly if we claim to have a deep respect for the Scriptures.