As the previous poster said, you can view his activity if he has been browsing Chrome while logged in from another device. However, you yourself would also need to be logged in to his Google account or Gmail to do so, and he may or may not get a notification that you are logged in.
You seem convinced that something bad is going on, and I know that feeling all too well... otherwise I would not have suggested a keylogger.
Are there other signs that he could be up to no good?
My ex had many Gmail and a hidden Yahoo. He used these to set up secret FB. Most of the hidden FB accounts I found seemed really odd, no friend or random "friends" from all over the place... he was using them to "keep an eye on me" (his words), but a couple of them were linked to dating sites. I knew it was impossible to have so many FB accounts (I found at least 7 fake ones) without him having either multiple phone numbers or multiple emails.
I managed to get a look at his phone. The only way I actually found evidence of his many Gmail accounts (used for Zoosk, Badoo etc. etc.), was because he had 10 or so text messages giving him a code for setting up the new account on his phone or for when he had forgotten his password for a few of the accounts (Gmail does this for security, you get a 5 or 6 digit code sent to your mobile which you then enter in Gmail to confirm it is in fact you in the account).
So I hear you on this. In my case though, there were small to medium sized red flags... But I did see a glimpse of something odd.... and he even gave me his phone to help him get rid of a virus, although he had cleaned off the dating apps but had left a fake FB page open, one which was linked to Zoosk and which he would use whenever we had an argument. He was trawling for women when we had a disagreement. But that is not the worst of it. He was abusive - and all the hidden online stuff was only one of the factors that led me to leave him.
It would have been illegal to put spyware on his phone. But it is not illegal to install it on your laptop etc., even if he is using it. If you really think something odd is going on and have other reasons to believe this, then that would a way of getting confirmation. Asking him, if he IS lying and hiding something, will only alert him to the fact that you are onto him, if he IS hiding something.
My ex denied, lied, made up stories about someone else using his phone, all sorts of stuff. He had no idea that I could view his Google activity. In the end, he WAS cheating and looking for sex clubs, all the while denying the existence of any other emails apart from the two he had and denying all the hidden FB. But my case is kind of extreme, although sadly does not seem to be all that uncommon since I joined here.
Leaving his phone unlocked shows in some way he has nothing to hide. However, my ex did the same... and had plenty to hide... just he had already deleted most of it and had no idea I would ever find it. But I didn't live with him so it was easier for him to deceive me. Clearing his history is odd. My ex cleared only the dodgy bits, but had no idea that they were all logged on Google Activity. So it "looked" innocent. But one of the things that alerted me was that he deleted kind of blocks of time when I KNEW we had been chatting on the Gmail messenger app, and that would be cleared too. When I looked in the Activity log, he had been searching for stuff he did not want me to find. And not nice surprises....
But proceed softly. I would say my gut instinct was spot on, all the time, and has never been wrong. I tried to ignore it but it kept nagging at me. But gently gently..... guns blazing and accusations is no way to do it.