Integrating contention-based random access procedures into low Earth orbit (LEO) satellite communication (SatCom) systems poses new challenges, including long propagation delays, large Doppler shifts, and a large number of simultaneous access attempts. These factors degrade the efficiency and responsiveness of conventional random access schemes, particularly in scenarios such as satellite-based internet of things and direct-to-device services. In this paper, we propose a deep learning-based random access framework designed for LEO SatCom systems. The framework incorporates an early preamble collision classifier that uses multi-antenna correlation features and a lightweight 1D convolutional neural network to estimate the number of collided users at the earliest stage. Based on this estimate, we introduce an opportunistic transmission scheme that balances access probability and resource efficiency to improve success rates and reduce delay. Simulation results under 3GPP-compliant LEO settings confirm that the proposed framework achieves higher access success probability, lower delay, better physical uplink shared channel utilization, and reduced computational complexity compared to existing schemes.