Comma or semicolon after "No" when responding to a question

Let's start with the interjection question. Interjections are punctuated with commas or exclamation points, which is explained at English Grammar Revolution:

Punctuating Interjections

Interjections are punctuated with an exclamation mark or a comma. Use an exclamation mark if the emotion is very strong; use a comma if the emotion is not as strong:

Wow! I won the lottery!
Wow, I have a peanut butter and jelly sandwich for lunch.

However, I don't believe you have an interjection in your sentence. As the same website says:

Beware!

Don't get fooled into thinking that all introductory words followed by an exclamation point or a comma are interjections - they're not!

Hmm... What is an interjection? Do you remember? It is a word that shows emotion. So, if the word in question does not show emotion, it is probably not an interjection.

Maria! Come and see the lion!

Names are not interjections because they do not show emotion. The tone of voice that you say them in may show emotion, but the name itself does not.

Stop! The lion will eat you!

The word stop is not an interjection. It is a verb because it shows action.

So, what part of speech is "No" in your sentence? To answer that, I'll point you toward Wikipedia, which claims that "Yes" and "No" are neither interjections nor adverbs, but parts of speech in their own right:

The words yes and no are not easily classified into any of the eight conventional parts of speech. Although sometimes classified as interjections, they do not qualify as such, and they are not adverbs. They are sometimes classified as a part of speech in their own right, sentence words, word sentences, or pro-sentences, although that category contains more than yes and no and not all linguists include them in their lists of sentence words.

That's just a brief excerpt; I strongly recommend you read the whole article. (The phrase "not all linguists" suggests a lack of consensus, and indicates this can be a thorny – and therefore interesting – problem.)

As for how to punctuate your sentence, I'd recommend a comma. Why do I say that? It looks more natural, it reads more natural, and that seems to be how I've seen it most often printed in books. A simple sample can verify that. If you click on that link, examine the search results, and use that as a guide, you'll find all of the following conventions used:

• No, the new software can still open the old files.
• No. The new software can still open the old files.
• No the new software can still open the old files.

but the first one (i.e., the comma) is by far the most prominent.

I agree with @NeilFein that the semicolon signals a longer pause, which in eye-reading tells the reader there's more to come; but in my use—which may be entirely idiolectal—a colon is called for, to tell the reader that what follows amplifies what went before.

No: the new software can still open the old files.

To my mind, a semicolon here is disjunctive and tells the reader that what follows will qualify what precedes.

No; but you will have to update your old files to take advantage of the new features.
No; and what is more, you will be able to identify many errors in your old files!

Both are correct; the semicolon simply indicates a longer pause after saying "No". The only difference in meaning would be that the comma version makes more sense when the reasoning is essential. If a "no" by itself would suffice to answer the question, the semicolon version would demonstrate that the reasoning is a little "tacked on". However, this change in meaning is extremely slight, and either would be correct in this case.