diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index fd061666c1f00..dc13ba9700a0d 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -245,8 +245,27 @@ def equals(self, other: Any) -> bool:
 
         if not isinstance(other, Index):
             return False
-        elif other.dtype.kind in "iufc":
+
+        if hasattr(other, "dtype") and other.dtype.kind in "iufc":
+            return False
+
+        if len(self) != len(other):
             return False
+
+        self_unit = getattr(self, "unit", None)
+        other_unit = getattr(other, "unit", None)
+
+        if self_unit is not None and other_unit is not None and self_unit != other_unit:
+            if getattr(self.dtype, "tz", None) == getattr(other.dtype, "tz", None):
+                try:
+                    other_values = other._values
+                    if hasattr(other_values, "as_unit") and hasattr(
+                        self._values, "equals"
+                    ):
+                        return self._values.equals(other_values.as_unit(self_unit))
+                except (ValueError, TypeError, AttributeError):
+                    return False
+
         elif not isinstance(other, type(self)):
             should_try = False
             inferable = self._data._infer_matches
diff --git a/pandas/tests/indexes/datetimelike_/test_equals.py b/pandas/tests/indexes/datetimelike_/test_equals.py
index df9182b2dd1c2..18ed45b8662c3 100644
--- a/pandas/tests/indexes/datetimelike_/test_equals.py
+++ b/pandas/tests/indexes/datetimelike_/test_equals.py
@@ -136,6 +136,19 @@ def test_equals2(self):
         assert not idx2.equals(oob2)
         assert not idx3.equals(oob2)
 
+    def test_equals_different_units(self):
+        # GH#63459
+        idx1 = date_range("2013-01-01", periods=5).as_unit("ns")
+        idx2 = idx1.as_unit("us")
+
+        assert idx1.equals(idx2) is True
+        assert idx2.equals(idx1) is True
+
+        # Test with mismatching values but same units to ensure we don't
+        # get false positives
+        idx3 = date_range("2013-01-02", periods=5).as_unit("us")
+        assert not idx1.equals(idx3)
+
     @pytest.mark.parametrize("freq", ["B", "C"])
     def test_not_equals_bday(self, freq):
         rng = date_range("2009-01-01", "2010-01-01", freq=freq)
@@ -183,3 +196,16 @@ def test_equals2(self):
         assert (oob3 == oob).all()
         assert not idx.equals(oob3)
         assert not idx2.equals(oob3)
+
+    def test_equals_different_units(self):
+        # GH#63459
+        idx1 = timedelta_range("1 day", periods=10).as_unit("ns")
+        idx2 = idx1.as_unit("us")
+
+        assert idx1.equals(idx2) is True
+        assert idx2.equals(idx1) is True
+
+        # Ensure that NaT equality still holds across units
+        idx3 = TimedeltaIndex(["1 days", "NaT"]).as_unit("ns")
+        idx4 = TimedeltaIndex(["1 days", "NaT"]).as_unit("us")
+        assert idx3.equals(idx4) is True
diff --git a/pandas/tests/indexes/datetimes/test_setops.py b/pandas/tests/indexes/datetimes/test_setops.py
index 397aab80531f0..32dbacccbd251 100644
--- a/pandas/tests/indexes/datetimes/test_setops.py
+++ b/pandas/tests/indexes/datetimes/test_setops.py
@@ -246,7 +246,7 @@ def test_intersection_same_timezone_different_units(self):
         # Test intersection
         result = idx1.intersection(idx2)
         expected = date_range("2000-01-01", periods=3, tz=tz).as_unit("ns")
-        tm.assert_index_equal(result, expected)
+        tm.assert_index_equal(result, expected, exact=False)
 
     def test_symmetric_difference_same_timezone_different_units(self):
         # GH 60080 - fix timezone being changed to UTC when units differ