Thanks for the update. I think I've got it mostly clear, except the pulse circuit. This schematic shows power going out to a cell pulse control circuit which is a separate item, right? So this will apply power steadily, and the pulse control circuit will look after pulsing the cell. is that right?
If you are not going to use relay's then this comment won't apply, but for the sake of my understanding anyway, please comment on this point...the way you show Relay1 doesn't make sense to me. If it is wired this way, the relay does nothing. Well, it switches on when the DC is applied, and it switches off when the DC is removed, but so what. If you remove Relay1 and just wire from F1 to S1 you get exactly the same operation. You say that it is a safety feature, but in fact it provides no safety at all as far as I can tell.
Similarly, I don't see the point of the second relay. If the Lower Water Level Sensor allows 12VDC to pass through to the relay, it will close, and 12VDC will flow through its contacts to the Cell Pulse Control Circuit. You would get exactly the same functionality if you simply remove the relay and wire from the Sensor to the Control Circuit. I can see one purpose for this relay, that is to "amplify" the current passing capability of the Lower Water Level Sensor so that enough current can flow to the Cell Pulse Control Circuit. But I suspect this isn't necessary. You can help determine this by providing a circuit or power requirement info for the Cell Pulse Control Circuit.
You have sprinkled rectifiers all over the place. I don't see why. For example, the rectifier near the coil of Relay1 is not necessary if you hook up the 12VDC the right way round. If you hook up the 12VDC the wrong way around, this rectifier prevents current flowing through the relay coil, but it doesn't prevent current flowing to other parts like to S1. It would be simpler to put one single rectifier in series with F1 to prevent all problems with reverse DC hookups. If you do this, you don't need the rectifier at the second relay, you don't need the rectifier on the Pulse Control Circuit ground return, you don't need the rectifier on the FLow Control Valve return and you don't need any of the rectifiers at either of the two connectors. You also don't need the rectifiers at the LEDs since the LEDs are in fact rectifiers.
Now, to get on to your interest in using an IC instead of the relays. It will probably be necessary to have some sort of relay or IC switch driving the water pump and control valve, but we need to know how much power they need. Please provide this info. One kind of switch that will work well is a high side switch, for example this one:
https://www.electro-tech-online.com/custompdfs/2007/07/FDC6330L.pdf
or a solid state DC switch, like this one:
https://www.electro-tech-online.com/custompdfs/2007/07/G3CN.pdf
I am interested to know what the circuit is connected to the Lower Water Level Sensor and the Upper Water Level Sensor. Can you describe these?