Comma Splice? "I'm curious, what is the time?"

Like others suggested, a dash (not a hyphen, be careful) or a colon are the most appropriated choices, but I'd exclude the comma. The examples would be as follows:

• I am curious — what is the time?
• I am curious: what is the time?

The comma must be excluded because it would constitute a comma splice, since it must be used along with a coordinator (and, but, not, so, etc). A semicolon can be adopted but sometimes it requires a conjunctive adverb at the beginning of the second clause (see hence, for example), so it depends on the case.

You don't have to end the sentence with a period; in this case you wouldn't. A period separates completely two ideas and in your example they are related so a period would not be advisable.

What's sure is that you can't avoid to use something. If you don't put anything between the two clauses, you'll end up writing a run-on sentence. In order to make it clearer about it, look at the example below, which shows a run-on sentence. It doesn't make much sense:

I am curious what is the time?

You can understand its meaning of course, but considering syntax rules, it's grammatically incorrect.