Learn French: Il me semble que ce document démontre un fait concret. - It seems to me that this document demonstrates a concrete fact.

This is the core of your current lesson! The phrase il me semble que (it seems to me that) expresses a personal opinion or conviction, which is treated as a certainty in your own mind. Therefore, it takes the indicative. If it were just il semble que (it seems that), it would express general doubt and require the subjunctive.

Since démontre is an -er verb, doesn't it look exactly the same in the subjunctive anyway?

Exactly. For regular -er verbs, the third-person singular form is démontre in both the present indicative and the present subjunctive. We only know it is grammatically indicative here because the rule for il me semble que demands it. If the verb were faire, it would clearly be the indicative fait rather than the subjunctive fasse.

Could I say ça me semble que instead of il me semble que?

No, il me semble que is a fixed impersonal expression, so you must use the impersonal pronoun il. While you can say ça me semble (that seems to me...) when describing a noun or adjective directly, you cannot mix the two structures into ça me semble que.

Is concret a false friend here?