I do not know what's going on here, but workaround is:
F[s_?NumericQ] := Cos[Sqrt[s]];
G[x_?NumericQ, y_?NumericQ] := F[Sqrt[x + I y]]
Plot3D[Abs[G[x, y]], {x, -1/10, 1/10}, {y, -1/100, 1/100}]
Or another way using ListPlot3D:
ClearAll["Global`*"]; Remove["Global`*"];
F[s_] := Cos[Sqrt[s]];
G[x_, y_] := F[Sqrt[x + I y]]
ListPlot3D[Partition[Flatten[Table[{i, j, Abs[G[i, j]]}, {i, -1/10, 1/10, 1/500}, {j, -1/100, 1/100, 1/500}]], 3]]
Regards M.I.