diff --git a/doc/source/whatsnew/v1.4.0.rst b/doc/source/whatsnew/v1.4.0.rst
index af7706f624323..3d518de98a8a3 100644
--- a/doc/source/whatsnew/v1.4.0.rst
+++ b/doc/source/whatsnew/v1.4.0.rst
@@ -500,6 +500,7 @@ Indexing
 - Bug in :meth:`DataFrame.nlargest` and :meth:`Series.nlargest` where sorted result did not count indexes containing ``np.nan`` (:issue:`28984`)
 - Bug in indexing on a non-unique object-dtype :class:`Index` with an NA scalar (e.g. ``np.nan``) (:issue:`43711`)
 - Bug in :meth:`Series.__setitem__` with object dtype when setting an array with matching size and dtype='datetime64[ns]' or dtype='timedelta64[ns]' incorrectly converting the datetime/timedeltas to integers (:issue:`43868`)
+- Bug in :meth:`DataFrame.sort_index` where ``ignore_index=True`` was not being respected when the index was already sorted (:issue:`43591`)
 - Bug in :meth:`Index.get_indexer_non_unique` when index contains multiple ``np.datetime64("NaT")`` and ``np.timedelta64("NaT")`` (:issue:`43869`)
 -
 
diff --git a/pandas/core/generic.py b/pandas/core/generic.py
index 71450a1e63bf3..c3ad87082c8ed 100644
--- a/pandas/core/generic.py
+++ b/pandas/core/generic.py
@@ -4633,10 +4633,17 @@ def sort_index(
         )
 
         if indexer is None:
+            if inplace:
+                result = self
+            else:
+                result = self.copy()
+
+            if ignore_index:
+                result.index = default_index(len(self))
             if inplace:
                 return
             else:
-                return self.copy()
+                return result
 
         baxis = self._get_block_manager_axis(axis)
         new_data = self._mgr.take(indexer, axis=baxis, verify=False)
diff --git a/pandas/tests/frame/methods/test_sort_index.py b/pandas/tests/frame/methods/test_sort_index.py
index c1141f705acbc..71822628473f4 100644
--- a/pandas/tests/frame/methods/test_sort_index.py
+++ b/pandas/tests/frame/methods/test_sort_index.py
@@ -9,6 +9,7 @@
     Index,
     IntervalIndex,
     MultiIndex,
+    RangeIndex,
     Series,
     Timestamp,
 )
@@ -418,6 +419,24 @@ def test_sort_index_ignore_index(
         tm.assert_frame_equal(result_df, expected_df)
         tm.assert_frame_equal(df, DataFrame(original_dict, index=original_index))
 
+    @pytest.mark.parametrize("inplace", [True, False])
+    @pytest.mark.parametrize("ignore_index", [True, False])
+    def test_respect_ignore_index(self, inplace, ignore_index):
+        # GH 43591
+        df = DataFrame({"a": [1, 2, 3]}, index=RangeIndex(4, -1, -2))
+        result = df.sort_index(
+            ascending=False, ignore_index=ignore_index, inplace=inplace
+        )
+
+        if inplace:
+            result = df
+        if ignore_index:
+            expected = DataFrame({"a": [1, 2, 3]})
+        else:
+            expected = DataFrame({"a": [1, 2, 3]}, index=RangeIndex(4, -1, -2))
+
+        tm.assert_frame_equal(result, expected)
+
     @pytest.mark.parametrize("inplace", [True, False])
     @pytest.mark.parametrize(
         "original_dict, sorted_dict, ascending, ignore_index, output_index",