Will showing him that I care drive him away?

I was dating a guy for about a month. We went on really good dates and spent a lot of time together during the month and had sex and we just get along well. I wanted to see him next week but he said he's so busy with football which I do believe so he can’t see me as much and I just said to him it’s okay and not to worry, I also just asked him if he was ready for anything serious and he just said he’s not but he really would be if he wasn’t so busy and that me and him are just fun. I said that’s fine by me but realistically I was upset. I wanna message him asking how his football match went but will this drive him away if we are just ‘fun’ in his eyes? I like him and I do care. We haven’t spoken for a few days since he’s been so busy but I just wanna show I care.
Will showing him that I care drive him away?
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