If you are using Super Socializer or Sassy Social Share plugin for Social Sharing, you can follow the steps mentioned below to show Whatsapp icon only on mobile devices.

For Sassy Social Share plugin:

Add following CSS in Custom CSS option in Miscellaneous section at plugin options page in admin panel:
@media screen and (min-width:783px){.heateor_sss_whatsapp_count, .heateorSssWhatsappBackground{display:none!important}}
For Super Socializer plugin:

Add following CSS in Custom CSS option at Super Socializer > General Options page in admin panel:
@media screen and (min-width:783px){.the_champ_whatsapp_count,.theChampWhatsappBackground{display:none!important;}}
783px in above code is the maximum width of the mobile device screen to show the Whatsapp icon. Screens wider than this won’t show Whatsapp icon. You can change it according to your needs.