In the wireless communication industry, achieving gigabit-per-second data rates with low-profile, ultra-wideband (UWB) microstrip patch antennae poses a significant challenge. Conventional optimization algorithms, though effective, are often computationally expensive, particularly for complex antenna geometries with high degrees of freedom. There is an imperative need for new methodologies to address this challenge and revolutionize the antenna optimization process. Successful and timely development of antennas relies on the efficiency and computational speed of optimization algorithms, full-wave electromagnetic solvers, and the intuition of radio frequency engineers. To mitigate the dependence on complex and time-consuming processes, we propose an efficient machine learning (ML)-based antenna optimization methodology that minimizes optimization time by more than 90%. This paper aims to apply and study the performance of two specific ML models, the radial basis function (RBF), and the least squared regression (LSR) models, in the bandwidth optimization without increasing the aperture area of a hexagon-shaped fractal antenna. The hexagon-shaped fractal antenna was chosen for its UWB characteristics, low profile, and high degrees of freedom (10 adjustable parameters). The reflection coefficient response of a hexagon-shaped fractal antenna is predicted by the trained RBF and LSR models and further optimized by the genetic algorithm (GA). The proposed approach stands out among other notable works in this research domain, especially for UWB applications, by prioritizing the optimization of the mean of the reflection coefficient across the entire frequency range instead of solely targeting individual frequency points. The GA-based optimization using trained ML models has increased the bandwidth by 21.3% and reduced the computational time by 90% compared to conventional optimization without increasing the physical or electrical size of the antenna. Simulation and measurement results concurred with a maximum difference of 5%, demonstrating the efficacy of the ML approach for antenna optimization.