This paper studies secure transmissions in a bursty interference channel constructed by opportunistic user scheduling. To improve the physical layer security, we propose a signal-to-noise-plus-interference ratio (SINR) based opportunistic transmission scheme and mathematically analyze the ergodic secrecy rate per transmitter-receiver pair. In this scheme, a subset among $K$ transmitter-receiver pairs is selected opportunistically, but each transmitter determines its activation state based on limited feedback of the SINR from its designated receiver only, without any information from other transmitters. For comparison, we also analyze the ergodic secrecy rates per transmitter-receiver pair for two benchmark schemes: non-opportunistic transmission and random transmission. Using derived analytical expressions, the achievable ergodic secrecy rates per transmitter-receiver pair are obtained by optimizing the activation probability for each scheme. To solve the non-convex optimization problem, we first show that the objective function is a sum of quasi-concave functions. Then, leveraging this fact, we propose a two-level iterative algorithm based on the damped Newton method. Furthermore, asymptotic behaviors of the achievable ergodic secrecy rate per transmitter-receiver pair are analyzed to offer insights behind the mathematical expressions. Our analytical and numerical results exhibit the gain of the proposed SINR based opportunistic transmission scheme compared to the benchmark schemes.