Eligibility Criteria:
Inclusion Criteria (Part 1):
* Female participants must be non-pregnant, non-breast feeding, and of non-childbearing potential
* Has a Body Mass Index (BMI) \<=32 kg/m\^2
* Has a body weight \>= 50 kg and \<= 100 kg
* Has been judged to be in good health based on medical history, physical examination, vital sign measurements, electrocardiogram (ECG), and laboratory safety tests
* Non-smoker or has not used nicotine or nicotine-containing products for at least 3 months
Inclusion Criteria (Part 2):
* Has been diagnosed with ITP at least 3 months prior
* Female ITP participants must be non-pregnant, non-breast feeding, and either of 1) non-childbearing potential or 2) must have serum beta human chorionic gonadotropin (HCG) level consistent with a non-pregnant state, and agree to use acceptable contraception from pretrial period until 84 days postdose
* Has a BMI \<=36 kg/m\^2
* Has been judged to be in good health, other than ITP diagnosis, based on medical history, physical examination, vital sign measurements, ECG, and laboratory safety tests
Exclusion Criteria (Part 1):
* Has a history or clinically significant endocrine, gastrointestinal, cardiovascular, hematological, hepatic, immunological, renal, respiratory, genitourinary, or major neurological abnormalities or diseases
* Has a history of cancer (malignancy)
* Has a history of significant multiple and/or severe allergies or has had an anaphylactic reaction or significant intolerability to prescription or non-prescription drugs or food
* Is positive for hepatitis B surface antigen, hepatitis C antibodies, or human immunodeficiency virus (HIV)
* Has had major surgery or donated or lost 1 unit of blood in the 4 weeks prior
* Has participated in another investigational trial within 4 weeks (12 weeks for biologics)
* Has received a live virus vaccination within 42 days or plans to receive such while participating in the trial
* Is unable to refrain from or anticipates the use of any medication, including prescription and non-prescription drugs and herbal remedies from 2 weeks prior and for the duration of the trial
* Consumes greater than 3 glasses of alcoholic beverages per day
* Consumes greater than 6 servings of caffeine-containing beverages per day
* Is currently a regular user of any illicit drugs or has a history of drug and/or alcohol abuse within 3 months
* Has a history of ITP or other autoimmune disease
* Has an active infection that is clinically significant
Exclusion Criteria (Part 2):
* Has a comorbid and significant hematological or immunological disorder
* Has a history of significant multiple and/or severe allergies or has had an anaphylactic reaction or significant intolerability to prescription or non-prescription drugs or food
* Is positive for hepatitis B surface antigen, hepatitis C antibodies, or HIV
* Has had major surgery or donated or lost 1 unit of blood within 4 weeks
* Has participated in another investigational trial within 4 weeks (12 weeks for biologics), excluding prior participation in the current study
* Has a history of ITP unresponsive to intravenous immunoglobulin (IVIG)
* Has had systemic corticosteroid use within 1 month (with the exception of stable low dose oral corticosteroids)
* Has had systemic IVIG or other systemic immunomodulatory therapy, excluding MK-8723 administration in the current study, within 3 months
* Has received a thrombopoietin receptor antagonist within 3 months
* Is unable to refrain from using thrombopoietin receptor agonists and/or systemic immune modulatory medications throughout the study
* Has received a live virus vaccine within 42 days prior or plans to receive such during the trial
* Consumes greater than 3 alcoholic beverages per day
* Consumes greater than 6 servings of caffeine-containing beverages per day
* Is currently a regular user of any illicit drugs or has a history of drug and/or alcohol abuse within 3 months
* Has clinical evidence of bleeding or coagulopathy including petechial rash, easy bruising, or excessive gingival bleeding with routine dental hygiene
* Has an active infection that is clinically significant