Can I use existing #4 wire to supply a 15A outlet?
Yes you can put 4 awg wire on 15 amp breaker.
However with this size wire you will find nothing fits a #4 directly so you will need to get a splice.
splices that actually connect wires as large as 4 awg to wires as small as 14 awg  the part number is 971022 Morris.  I just happen to have a dozen on my desk.
you can find them for ~6$ ea . Depending on how far away the receptacle is this could be both a big savings and no voltage drop issues. Or just easier than pulling a new line but it is totally code compliant to do this.
There are other brands of these connectors / splices like Polaris I just happen to have the exact model you need on my desk (they are very handy).