Here's a one-liner that uses BEDOPS closest-features on a UCSC-derived refGene list of genes:

$ closest-features --closest --no-ref --no-overlaps --dist refGene.hg38.bed refGene.hg38.bed | cut -d'|' -f2 | grep -v NA | awk '{ if($1<=0){ $1*= -1;} print $1;}' | Rscript -e 'summary(as.numeric(read.table(file("stdin"))[,1]))'
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
      1    1185    6411   25654   22165 1687452

The median distance between genes is 6411nt. The mean is 25kb, etc.

The file refGene.hg38.bed is sorted with BEDOPS sort-bed.

If you're starting from GFF3, you can use BEDOPS gff2bed:

$ awk '($3 == "gene")' annotations.gff | gff2bed - > annotations.bed

To incorporate into the above one-liner, using bash process substitutions:

$ closest-features --closest --no-ref --no-overlaps --dist <(awk '($3 == "gene")' annotations.gff | gff2bed -) <(awk '($3 == "gene")' annotations.gff | gff2bed -) | cut -d'|' -f2 | grep -v NA | awk '{ if($1<=0){ $1*= -1;} print $1;}' | Rscript -e 'summary(as.numeric(read.table(file("stdin"))[,1]))'