"I'm doing business in Alameda right now. Will be back later tonight."

Can you say this:

"I'm doing business in Alameda right now. Will be back later tonight."

Would a native speaker say this?

You might hear it. What do you think might be wrong with it? You need to remember that when we speak, we don't consciously form grammatical sentences so there is a difference between what we might say and what we might write. There will also be a difference depending on our audience.

emsr2d2 said:

You might hear it. What do you think might be wrong with it? You need to remember that when we speak, we don't consciously form grammatical sentences so there is a difference between what we might say and what we might write. There will also be a difference depending on our audience.

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I don't think there's anything wrong with it, but maybe it's somewhat un-native like or you wouldn't hear a native speaker say it. Would it be better to say this:
"I'm doing business in Alameda right now, and will be back later tonight."

Would a text message - "I'm doing business in Alameda right now. Will be back later tonight." - be ungrammatical or unnatural?

Some people send entirely grammatically correct text messages. Some people are impatient and use textspeak. There are multiple possibilities, including:

I'm doing business in Alameda right now. I'll be back later tonight.
Doing business in Alameda right now. Be back later tonight.
In Alameda on business. Back tonight.
Biz in Alameda.Bck 2nite.

... and lots of other variations. There is no single right way to say/write most things in English. Again, it depends on the audience. My first example sentence above might be suitable for sending to your boss but the third one probably isn't. The fourth definitely isn't. In my opinion, that shouldn't be sent to anyone but I am not a fan of textspeak, no matter how much time it saves.

I am not a teacher.

I think, "Will be back later tonight" is the sort of thing you would expect to see written on a note tacked to the door. It's like, "Gone fishing".

This type of abbreviated not-entirely-grammatical message is quite natural, so I think it would also be acceptable in a text message.