According to credit bureau Experian, being denied a credit card will not cause your credit score to go down because you were rejected. The reason is that your credit report does not provide details on whether or not any application has been declined or approved. Since the credit bureaus do not know, a decline does not create an impact on a credit score.
Applying for the card does create a hard inquiry though. Such an inquiry demonstrates that you applied to obtain more debt even though the available credit or debt does not yet show up on your personal credit report. The fact that no new account appears on your report does not mean your application was denied. It alternatively could mean that you decided not to accept the credit account or loan.
Applying for a credit card (whether it is accepted or rejected) does harm your credit score in the 10 percent component for new credit inquiries. This results from the bureaus understanding that possible new debt creates potential credit risk. The good news is that such an effect is usually quite small. It will decline in importance with time too.
Remember that timely payment history on approved accounts (at a 35 percent of credit score component) is the most important element in your credit scoring.
To put this into perspective, this payment history outweighs the impact of new credit inquiries by three and one half times.