It is fine to send *a* text saying that you are there for your sufferer when and if they are ever ready to talk... But sending multiple messages is violating their boundaries. If somebody requests space or leaves, then sending multiple messages, even if they're spaced apart, is not respecting their wishes. That is no way to endear yourself to somebody you love. It demonstrates that you are more worried about your own feelings than theirs.
Honestly, in this situation, there is not much you can do except take care of yourself.