Can a candidate for US Army enlist after having varicocele surgery?

Yes, IF your surgery is successful, then you are likely to be accepted. However, note that your varicocele can redevelop after surgery if you don't maintain lifestyle improvements. There are many risk factors for varicocele, for example, exercising can be a big aggravating risk factor. So, you must ensure to exercise the right way in order to minimize any stress on the varicocele, and maximize the therapeutic benefits. Learn more about how to exercise with varicocele. Other varicocele risk factors include, but are not limited to:

• Diet
• Masturbation
• Bowel / Digestive Health
• Exercise Type
• Excessive Sitting (6+ hours/day)
• Posture
• Hip tightness & mobility
• Underwear (Big Risk Factor)
• and more...

Learn more about varicocele lifestyle risk factors.

Did you know that you can also perform treatments to improve post-surgery results?

• Varicocele Health Kit: Herbal supplements can help reduce damage to vein valves and improve venous blood drainage. They also have been demonstrated to improve post-surgery results, and prevent varicocele grade from increasing.
• Yoga Inversions: Nothing beats inversions for recover and draining the stagnant pooled toxic blood in the testicular veins! Inversions have been long-used in sports and performance activities because of their effects on enhancing recovery. Turns out they help with varicocele too!
• Cooling Treatments: Now, don't perform cooling treatments post-surgery without your doctor's consent, but, you can always make it your goal to normalize testicular temperature. This helps prevent the veins from swelling and keeps your testicles functioning normally! 
• and more...

I hope that serves as a good introduction to what exactly what you need to know about enlisting in the army, and what your options are!