Is this considered ‘cheating’ and who is in the wrong?

Many partner left his phone unlocked and I had a look at his WhatsApp messages. I found out he is texting his ex partner who we saw in the supermarket the other week. It was a conversation asking how she is and that it was nice to see her. When I asked him if he has been messaging her since we’ve seen her he denied it. I then accused him of lying and he asked me why I would even ask that in the first place. I then said I could see in his eyes he was lying not mentioning I had a look at him phone. We ended up arguing all evening and now he wants to end it with me. We have been living together 2 years and have a daughter. I don’t know if I can trust him anymore with anything. Should I agree with ending the relationship or just give it a break for a while?
Is this considered ‘cheating’ and who is in the wrong?
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