Step 1: Understanding the Concept
This question tests the use of modals in a conditional sentence. The word "Unless" introduces a condition.
Step 2: Key Concept
"Unless" means "if ... not". The sentence structure is a Type 1 Conditional, which describes a real or likely situation in the future.
The structure is: Unless + present simple, ... will + base form of the verb.
So, the sentence can be rephrased as: "If he does not beg my pardon, I will not excuse him."
Step 3: Detailed Explanation
The first part of the sentence, "Unless he begs my pardon," is in the present simple tense.
According to the rule for Type 1 conditionals, the main clause should use the future simple tense, which is formed with "will" + verb.
Therefore, the blank should be filled with "will".
The sentence becomes: "Unless he begs my pardon, I will not excuse him."
Step 4: Final Answer
The sentence is a first conditional, requiring "will" in the main clause to indicate a future result.