I told one lie to my boyfriend and I need to fix it

I have been with my boyfriend for 1 yr and 3 months. We have recently moved in together (about 2 months now). Back in July of 2019 we got in a car accident. He was driving and thankfully no one was hurt. He did however get a DUI. Since then I have been driving him to work at 5 am and picking him up at 4:30 pm. Back in August I was 3 hours late picking him up. I lied about why I was late. I said I was studying in the library and time got away from me. I am in college so I have planned my work and classes around his schedule just so I could be there to pick him up in the afternoons. Well the day I was late I was actually meeting up with 2 of my friends that are women that I hadn't seen in a while. One of them had a baby and had a busy schedule so I didn't want to reschedule just because I had to pick my bf up from work. He said since that day he thought I cheated but I didn't. I told him the truth about a month ago about meeting up with my friends and he said that I did a very piece of shit thing. That how can he trust anything I say. I have only ever told that one lie. I have been a faithful and trustworthy gf with the exception of that one lie. He has asked me not to talk to him and I told him I would really like to talk when he is ready and to let me know when. I have given him space for 4 days now and I want to talk things out. I want to show him the texts to my friends that prove I am telling the truth now and explain to him that I felt under appreciated being his personal driver. I shouldn't have lied I should have just told him I couldn't pick him up that day that I wanted to go see my friends and catch up. I obviously love him and want to be with him otherwise I wouldn't still be driving him back and forth to work. Should I give him more space or should I ask if we can talk it out? Is there any way to fix this?

You guys need to understand that part of the success of your relationship will be you pair sharing values and standards. You obviously share goals as you have moved in together. Way back in August 19 you were late to pick up your man from work and he immediately thought that you were cheating on him. Now that he knows the truth about it, he's deemed it just as bad. So he had trust and insecurity issues then without your lie adding to it. To restore your trust you need to give him space and you need to communicate with him when he's ready to..but first you need to determine where your relationship is going. He'll come back to you when he's ready and it'll be his actions which will tell you where you stand with him. You guys need predictability within your relationship together and it's not so much about you being a taxi for him when his actions alone cost his license in the first place, it's about you both being upfront and honest with each other which will go along way to restoring the trust that your relationship needs to survive.