What I love about this scene is how it shows how much Chief cares for Cortana. He’s literally spent the most part of his life following orders from his superiors - doing everything they say, making sure they were content with his work and his outcome. He’s always done his best - not necessarily wanting to please but wanting to get the job done. He was built to follow orders and do his job - taught from a young age that that was the way of life and the only way. But when Cortana’s life comes into the question, those years of sticking to the regulations and saying, “Yes, ma’am/sir” every single time he’s given an order goes out the window. Cortana is more important to him than his whole purpose in life.
His prime directive was to protect her at all costs as well as humanity. To surrender her would not only negate his purpose but also to risk humanity as a whole, but we all know he would sacrifice it all for her. She was his partner, his voice of reason. I’d they were going anywhere it was together, either into eternity or the abyss. The problem is at the end of the day he will fade into time and pass away and she will be left alone. Hence her willingness to sacrifice herself so readily. The thought of being alone with john literally kills her.