readonly List<TextNode> _spellCheckNodes = new List<TextNode>();

private void SpellCheck()
{
 _spellCheckNodes.Clear();

 AddTextNodes(richTextEditor.Document.RootNode.ChildNodes, _spellCheckNodes);

 AddBindingsToSpell();
}

private void AddTextNodes(NodeCollection nodes, IList<TextNode> textNodes)
{
 foreach (var node in nodes)
 {
  if (node is TextNode)
  {
   textNodes.Add(node);
  }
     
  if(node.HasChildNodes)
   AddTextNodes(node.ChildNodes, textNodes);
 }
}

private void AddBindingsToSpell()
{
 spellChecker.SpellCheckTargets.Clear();

 foreach (TextNode node in _spellCheckNodes)
 {
  Binding bind = new Binding("Text")
  {
   Source = node,
   Mode = BindingMode.TwoWay
  };

  spellChecker.SpellCheckTargets.Add(bind);
 }

 spellChecker.SpellCheck();
}

Rich text document parts are stored as TextNodes. If I applied different style to the part of some word, it would appear in two different textnodes. Thus it is not possible to correctly check spelling of the word.