I was installing a new ceiling light fixture in my home. I attached all of the wires and was screwing in the last bolt for the mount when I heard a loud popping sound and saw a bit of smoke. The breaker tripped, making me realize that I didn't turn off the breaker.
After switching on the breaker and putting in a lightbulb, I found that everything works perfectly. However, I am worried that there could be a dead short or some other issue that could cause a fire. Should I have a professional come by to check it out?

Remove fixture and check for exposed wire or wires damage by mounting screw.

If you look carefully you may find the carbon smudge caused by the *pop*... if you do look closely to see if the screw or wire appears to be damaged. Then maybe you can decide whether some tape or a new fixture is required...
If you cannot identify the source of the issue a look by a trusted pro can help you sleep better... good luck

I was looking at the wires I can not find any blackened areas. Should I just wrap the wires with electrical tape or have an electrician investigate this further?

That decision is yours Mike... I would like to know where the *pop* came from
Before I walked away from this.