Connected Vehicles (CVs) can leverage the unique features of 5G and future 6G/NextG networks to enhance Intelligent Transportation System (ITS) services. However, even with advancements in cellular network generations, CV applications may experience communication interruptions in high-mobility scenarios due to frequent changes of serving base station, also known as handovers (HOs). This paper proposes the adoption of Open Radio Access Network (Open RAN/O-RAN) and deep learning models for decision-making to prevent Quality of Service (QoS) degradation due to HOs and to ensure the timely connectivity needed for CV services. The solution utilizes the O-RAN Software Community (OSC), an open-source O-RAN platform developed by the collaboration between the O-RAN Alliance and Linux Foundation, to develop xApps that are executed in the near-Real-Time RIC of OSC. To demonstrate the proposal's effectiveness, an integrated framework combining the OMNeT++ simulator and OSC was created. Evaluations used real-world datasets in urban application scenarios, such as video streaming transmission and over-the-air (OTA) updates. Results indicate that the proposal achieved superior performance and reduced latency compared to the standard 3GPP HO procedure.