Select Pairs in a List

I think you'll find the simple

DeleteCases[list, {_, _, __}, {3}]

to be a bit more efficient...

Using ReplaceAll (/.)

list /.  {_Integer, _Integer, __} :> Nothing

One needs to carefully specify the replacement rules however, try for example

list /.  {_, _, __} :> Nothing
(* Nothing *)

Rule replacement is actually quite efficient (relatively speaking) in this case, compare the above to @ciao's answer:

testList = Join[Apply[Sequence]@Table[list, 200000]]

(res1 = testList /.  {_Integer, _Integer, __} :> Nothing); // AbsoluteTiming
(* {2.55462, Null} *)
(res2 = DeleteCases[testList, {_, _, __}, {3}]); // AbsoluteTiming
(* {0.769297, Null} *)
res1 == res2
(* True *)

For a solution based on rule replacement a factor of ~3 worse than the (supposedly) fastest method isn't half bad.

Using Apply

f[x_, y_] := {x, y};
f[___] := Nothing;
Apply[f, list, {3}]