To have integrity is to live rightly. "Rightly" means, in alignment with your own Divine heart. That's true integrity.
Integrity supports your relationship with God, and your ability to be/feel close to God. If you're out of integrity, you may not feel good enough or strong enough to be presentable before God. Also, God cannot help or guide someone who persists in ignoring what they know is right, true, and what their heart wants them to do.
A life of real integrity is naturally attuned to God. After all, what God wants is the same as what you want. It is God's heart that lives in you. That's why you love beauty, truth, and rightness.
When you go against God's will, you are going against your own will -- like a woman in love, who thinks she shouldn't be in love, so she suppresses and invalidates her own desire. She's using her will to counteract her true feelings -- her true will. Will against will, hope against hope.
In the same way, people live in all kinds of ways that they hate, doing things they don't like, things they feel bad about. But meanwhile, and always, to please God is to please yourself. If you do what God wants, you're doing what you want. And if you refuse what God wants, then you're refusing your own heart.
That's why, to be at integrity is to be one with God, in will and in vision. Now you know exactly what God has to do with integrity.
Such is the connection between God and Integrity.