It sounds like you need a Python program that checks whether two numbers are positive integers, performs the Euclidean division on them to find the greatest common divisor (GCD), and returns the GCD according to the described division process. Here's a Python implementation that does precisely that:

```python
def gcd(a, b):
    """Compute the greatest common divisor of a and b using the Euclidean algorithm."""
    # Ensure a and b are positive integers
    if not (isinstance(a, int) and isinstance(b, int)):
        return "Both numbers must be integers."
    
    if a <= 0 or b <= 0:
        return "Both numbers must be positive."
    
    # Assign a and b such that a >= b
    if a < b:
        a, b = b, a
    
    # Perform Euclidean division until remainder is 0
    while b != 0:
        a, b = b, a % b
    
    return a

# Get input from the user
def main():
    try:
        num1 = int(input("Enter the first positive integer: "))
        num2 = int(input("Enter the second positive integer: "))
        result = gcd(num1, num2)
        print("The GCD is:", result)
    except ValueError:
        print("Please enter valid integers.")

if __name__ == "__main__":
    main()
```

### Explanation:

1. **Function `gcd(a, b)`**:
   - This function computes the GCD of two integers `a` and `b` using the Euclidean algorithm.
   - First, it checks if the inputs are integers and are positive.
   - It then ensures that `a` is the larger number.
   - It iteratively updates values of `a` and `b` (where `b` is set to the remainder of dividing `a` by `b`) until `b` becomes zero.
   - It returns the last non-zero value of `b` (stored in `a` after swapping in the loop), which is the GCD.
   
2. **`main()` function**:
   - This function prompts the user for two integers.
   - It handles potential input errors using a try-except block for conversion to integer.
   - It calls the `gcd()` function and prints the result.

The code ensures that the inputs are valid positive integers and uses a robust method (Euclidean algorithm) for calculating the GCD. The script gracefully handles invalid input by displaying appropriate messages.