To solve the given optimization problem using Gurobi, we first need to translate the natural language description into a mathematical formulation. The objective is to maximize the function: $2.04 \times \text{milligrams of vitamin C} + 8.38 \times \text{milligrams of vitamin B4}$.

Given constraints:
- Immune support index for vitamin C: 0.24
- Kidney support index for vitamin C: 0.03
- Immune support index for vitamin B4: 0.73
- Kidney support index for vitamin B4: 0.9
- Total immune support index should be at least 20 and at most 36.
- Total kidney support index should be at least 27 and at most 51.
- $4 \times \text{milligrams of vitamin C} - 2 \times \text{milligrams of vitamin B4} \geq 0$.
- Vitamin C must be a non-fractional amount, but vitamin B4 can be fractional.

Let's denote:
- $x_0$ as the milligrams of vitamin C,
- $x_1$ as the milligrams of vitamin B4.

The objective function is: $\max 2.04x_0 + 8.38x_1$

Constraints:
1. $0.24x_0 + 0.73x_1 \geq 20$
2. $0.24x_0 + 0.73x_1 \leq 36$
3. $0.03x_0 + 0.9x_1 \geq 27$
4. $0.03x_0 + 0.9x_1 \leq 51$
5. $4x_0 - 2x_1 \geq 0$

Now, let's write the Gurobi code for this problem:

```python
from gurobipy import *

# Create a new model
m = Model("Vitamin_Optimization")

# Define variables
x0 = m.addVar(vtype=GRB.INTEGER, name="milligrams_of_vitamin_C")  # Non-fractional amount of vitamin C
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="milligrams_of_vitamin_B4")  # Fractional amount allowed for vitamin B4

# Define the objective function
m.setObjective(2.04*x0 + 8.38*x1, GRB.MAXIMIZE)

# Add constraints
m.addConstr(0.24*x0 + 0.73*x1 >= 20, name="immune_support_min")
m.addConstr(0.24*x0 + 0.73*x1 <= 36, name="immune_support_max")
m.addConstr(0.03*x0 + 0.9*x1 >= 27, name="kidney_support_min")
m.addConstr(0.03*x0 + 0.9*x1 <= 51, name="kidney_support_max")
m.addConstr(4*x0 - 2*x1 >= 0, name="vitamin_ratio")

# Optimize the model
m.optimize()

# Print the results
for v in m.getVars():
    print(f"{v.varName}: {v.x}")
print("Objective:", m.objVal)
```