diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 67456096e8681..bccc2e0c52568 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -4444,8 +4444,8 @@ def asof_locs(self, where, mask):
 
     def sort_values(
         self,
-        return_indexer=False,
-        ascending=True,
+        return_indexer: bool = False,
+        ascending: bool = True,
         na_position: str_t = "last",
         key: Optional[Callable] = None,
     ):
@@ -4509,7 +4509,9 @@ def sort_values(
 
         # GH 35584. Sort missing values according to na_position kwarg
         # ignore na_position for MutiIndex
-        if not isinstance(self, ABCMultiIndex):
+        if not isinstance(
+            self, (ABCMultiIndex, ABCDatetimeIndex, ABCTimedeltaIndex, ABCPeriodIndex)
+        ):
             _as = nargsort(
                 items=idx, ascending=ascending, na_position=na_position, key=key
             )
diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index 54c8ed60b6097..7f9e39257bc8e 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -41,7 +41,6 @@
 )
 from pandas.core.indexes.numeric import Int64Index
 from pandas.core.ops import get_op_result_name
-from pandas.core.sorting import ensure_key_mapped
 from pandas.core.tools.timedeltas import to_timedelta
 
 _index_doc_kwargs = dict(ibase._index_doc_kwargs)
@@ -164,22 +163,6 @@ def __contains__(self, key: Any) -> bool:
             is_scalar(res) or isinstance(res, slice) or (is_list_like(res) and len(res))
         )
 
-    def sort_values(self, return_indexer=False, ascending=True, key=None):
-        """
-        Return sorted copy of Index.
-        """
-        idx = ensure_key_mapped(self, key)
-
-        _as = idx.argsort()
-        if not ascending:
-            _as = _as[::-1]
-        sorted_index = self.take(_as)
-
-        if return_indexer:
-            return sorted_index, _as
-        else:
-            return sorted_index
-
     @Appender(_index_shared_docs["take"] % _index_doc_kwargs)
     def take(self, indices, axis=0, allow_fill=True, fill_value=None, **kwargs):
         nv.validate_take(tuple(), kwargs)