How can I store asymmetric keys server-side, so that they are only readable by devices? I am building a platform in which there is a central server &amp; devices (computers).

These computers hold the same keys pairs that correspond to specific resources (shared between devices), so that only these devices can return signed data if required.

In that setup my question is:

How to store keys on the server-side, with only devices being able to read them AND no device interactions to replicate those keys?
Problem Statement:
I will answer this question as I think I understand your problem. I will write down what I've understood from your problem:

Your system is comprised of a central server and some nodes/devices/computers.
You need a digital signature of the devices as well as the server tying them to non-repudiation constraint (They can't deny having taken action.).
The public and private key pairs between devices shouldn't be exchanged between them but rather setup by the server.
The devices should be able to communicate with each other using those asymmetric key pair.

Assumptions:

You can't change packet format or constitution.
Every device obtains its asymmetric key pair from the server.

Solution:
Let every device have a global unique identifier (e.g Medium Access Control address), the configuration of the server takes place like this:

The server opens itself to requests (time-slotted) and creates a table of all the devices (IP, MAC, parameters based on design) then creates 2n keys (Public and Private) for n devices and communicates it to them.
Any device A which intends to communicate to any other device B requests B's public key from the server and locks the message using its (A's) private key.
The message is then passed on to the server which identifies the communicating parties A, B and passes it on to B.
B reads the message by unlocking it using its private key and finds it digitally signed using A's public key.

I would have liked to comment and verify a few details to make this answer  better catered to your problem but I need to gain some rep before commenting.
There are several security issues which will still remain but I will only solve them when you care enough about them.