Choose the most appropriate option to fill in the blank.No nation can…

The correct answer is Option (2) → until

Conjunction "until" after a negative:

• Until means up to the time when. After a negative clause it means that something will not happen before a condition is met.
• Clue word: the negative "No nation can progress".
• This sentence: "No nation can progress" (negative) until it adopts a fair and just approach: progress comes only after the welfare of citizens is cared for.
• Complete sentence: No nation can progress until it adopts a fair and just approach towards the welfare of its citizens.
  Example: You cannot go out to play until you finish your homework.

Why the Other Options are Incorrect?

Option 1: though — Incorrect. Though shows contrast (in spite of the fact that); it would say that no nation progresses in spite of being fair, which is illogical.
Example: Though he was ill, he came to school.

Option 3: since — Incorrect. Since gives a reason (because); it would say that a fair approach stops a nation from progressing.
Example: Since it was late, we took a taxi.

Option 4: when — Incorrect. When gives the time of an action; "No nation can progress when it adopts a fair approach" says the opposite of what the sentence means.
Example: Call me when you reach home.

Contrast: "until" after a negative names the condition that must come first; "though" shows contrast, "since" gives a reason, and "when" gives a time.