Equivalent of RadialOutside for Graph VertexLabels

g = Graph[CompleteGraph[26], 
VertexLabels -> Table[i -> Placed["Name", {{0,0}, 
{-Cos[Pi/2 + 2 i Pi/26], .25 - Sin[Pi/2 + 2 i Pi/26]}}], {i, 26}]]

Not to detract from PatoCriollo's excellent answer, but just to show that there is always a "there is also...".

Furthermore, the following, to my surprise, is not as fragile as I thought it might be with respect changes in ImageSize and in the vertex count of CompleteGraph.

vc = GraphEmbedding[CompleteGraph[26]];
g = Graph[EdgeList@CompleteGraph[26],
  VertexLabelStyle -> Directive[{16, Bold, "Panel"}], ImagePadding -> 20,
  VertexLabels -> Table[i -> Placed["Name", .5 + Pi vc[[i]]], {i, 26}]]

gr = With[{vc = GraphEmbedding[CompleteGraph[#]]},
    Graph[EdgeList@CompleteGraph[#],
     VertexLabelStyle -> Directive[{16, Bold, "Panel"}], 
     ImagePadding -> 20, ImageSize -> #2,
     VertexLabels -> Table[i -> Placed["Name", .5 + Pi vc[[i]]], {i, #}]]] &;

Row[gr[#, 300] & /@ {10, 16, 26}]

Row[gr[16, #] & /@ {200, 300, 500}]

If you don't mind having a Graphics rather than a Graph object, you can use GraphComputation`GraphPropertyChart which combines a PieChart with the edges of a graph:

GraphComputation`GraphPropertyChart[CompleteGraph[16], 
 ChartStyle -> "Rainbow", 
 ChartBaseStyle -> Directive[EdgeForm[], Opacity[.3]]]

Use the options LabelingFunction -> (Placed[#2[[2]], "VerticalCallout"] &) and ChartLabels -> None to get

Use CompleteGraph[32] as the first argument to get

To remove the annuli, use ChartElementFunction -> None:

 GraphComputation`GraphPropertyChart[CompleteGraph[25], 
  ChartStyle -> "Rainbow", 
  ChartLabels -> Placed[Range[25], "RadialCallout"], 
  ImageSize -> 450, ChartElementFunction -> None, 
  ChartBaseStyle -> Opacity[.5]] /.  _Disk :> {}

Use ChartLabels -> Placed[Range[25], "VerticalCallout"] to get