I have a switch that currently operates a single outlet. I would like to install a ceiling fan operated by the switch. I cut the line between the switch and outlet, rendering that line powerless. If I install a junction box and bring the wire from the outlet, the switch, and the fan together will the fan and outlet both be powered and operated by the switch?

Why did you cut the wire??
You could have used that wire to convert it to feed the switch. Then just run a new line to the fan.

Well, I have a new issue. I decided to put everything back together as it was before I messed with it. When I was re-joining the cut wire (running between switch and outlet), I added a "branch" to power the fan. The strange thing is, when I test the connections with my voltage indicator, I get these results: hot + ground: positive test. Neutral + ground: positive test. Hot + neutral: nothing. Shouldn't the voltage test light go on when you touch the hot and neutral wires?

First of all, how did you "add a branch"? Did you add a box somewhere?

Second, you are dealing with a switch loop. This is where the white is a hot feeding the switch and the black is the return to the light or switched receptacle. You cannot take constant power from that line.
You can though convert it to a constant feed, but you will lose the receptacle being switched, it will not be hot all the time.