What established lore? I have read every book and played every game many times over. It's not there. What you're describing is beyond John's character.
If you don't think the Chief and Cortana have developed an emotional attachment beyond friendship over the course of the Halo games, that's fine. No point arguing further, as even though it is established lore, if you don't see anything there then there's nothing I can say to sway your opinion.
Besides, Frankie already said no romance is happening.