Fragile watermarking is one of the integrity protection mechanisms of multimedia data like image, video, and audio. A secure fragile image watermarking scheme, which is used to detect modification or tampering occurred on the image content, is devised in this paper. The projected scheme is developed into two steps, i.e., computation of secure authentication code/watermark bit from some of the most significant bits of each pixel, and subsequently hides the watermark bit in the least significant bit (LSB) of each pixel by suggested watermark embedding procedure. The watermark bit is concealed into the LSB position of each pixel by a secret sequence obtained from a logistic map. This procedure boosts the security since without knowing the secret sequence, the authentication code extraction from each pixel is not possible. The suggested watermarking approach is assessed on a set of grayscale images, and a decent quality of the watermarked image is visualized. After performing several malicious tampering attacks on the watermarked images, the results of the experiment depict that the tamper detection procedure can effectively locate or identify the altered region from the watermarked image. The scheme is also evaluated based on different estimated quantity parameters like PSNR, FPR, FNR, and Image Fidelity, and satisfactory results are attained.