2004 Lincoln navigator air suspension all the way down front and back if I was to jump the relay to turn compressor on should the truck start raising up unless compressor not pumping or a leak

Ok do you here the compressor running if the truck is all the way down the sensors should be telling the computer that and it should constantly keep trying to raise the suspension if you don't hear it check the fuses and relays at the power distribution box if you do hear it the orings seals are bad and the spring farthest away will fill up first unless it's leaking if it is the truck will not come up to check inspect the bags for tares in them if none and the compressor is cycling go from the front to the back spraying soapy water at the lines if no bubbles now go to the air springs and do the same thing the bags first then the solenoid valves that fit into the top of the air bag so you know whenever these vehicles are worked on and the bags are deflated you must change the o rings seals everytime and if you put to much wieght in the truck with people or cargo you blow them out ok if you jack up the truck to take wieght of the springs they will take some air and that's when you spray the soapy solution as you release the jack the pressure on the bag will increase forcing air out of the weak spot which will show as a bubble ok good luck stephen

Yea definitely sounds like computer failure. For about half the price of just the new computer you can put a brand new coil suspension on that Navigator and never worry about it again. These Navigators and the Expeditions from that same year range all had awful issues with the air suspension. Check these guys out:
2003 2006 Lincoln Navigator 4 Wheel Standard Air Suspension Conversion Kit...

No it wont. the air suspension works as a complete system. each air spring has an electric solenoid, each corner has an electric height sensor and then the compressor itself. just energizing the compressor will not open the solenoids or sensors both of which are needed to raise the vehicle. what is probably happening is a failure in the computing system for the suspension system. expensive to repair.