Is it acceptable for professors to send letter of recommendation via a personal email?
I would not worry about this

Professors using personal email is not uncommon.  I have encountered this multiple times in the past.  In academia, there are fewer standards of professionalism and is a lot more freedom for people to do what they want (compared to the business world, for example, where using personal email would typically be frowned upon).  A professor using personal email for work communication just isn't that unusual.

The sender's email address is not a good way to verify identity, anyway.  Nobody should be relying on this as a way to verify that something really came from a sender, because your professor's work email could easily be spoofed, with no completely reliable way to detect it.  Here is one discussion of that on Security Stack Exchange.  Of course, even if it is not reliable someone still might do it.  But these days, practically everyone has experience of spam that appeared to come from someone they know.  Hopefully people are not using the "from" field on the email to verify anything important.

If anyone did suspect fraud, the logical response would be to check.  In this case, that would mean getting in touch with the professor and discovering the letter really was from him.  It's exceedingly unlikely that they would discard your application based on the mere suspicion of fraud, without any real evidence (and doing so would be extreme incompetence on their part).