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Preventing Veteran suicide is a national priority. The US Department of Veterans Affairs (VA) collects, analyzes, and publishes data to inform suicide prevention strategies. Current approaches for detecting suicidal ideation mostly rely on patient self report which are inadequate and time consuming. In this research study, our goal was to automate suicidal ideation detection from acoustic and linguistic features of an individual's speech using machine learning (ML) algorithms. Using voice data collected from Veterans enrolled in a large interventional study on Gulf War Illness at the Washington DC VA Medical Center, we conducted an evaluation of the performance of different ML approaches in achieving our objective. By fitting both classical ML and deep learning models to the dataset, we identified the algorithms that were most effective for each feature set. Among classical machine learning algorithms, the Support Vector Machine (SVM) trained on acoustic features performed best in classifying suicidal Veterans. Among deep learning methods, the Convolutional Neural Network (CNN) trained on the linguistic features performed best. Our study shows that speech analysis in a machine learning pipeline is a promising approach for detecting suicidality among Veterans.